Commander Jon Savell from the Metropolitan Police said the breakdown of the relationship could be a motive for the attack last week.
Police have warned that the facial injuries suffered by Abdul Ezedi, 35, in the attack in Clapham, south London, may be fatal if left untreated.
Officers will return to the scene of the attack on Wednesday evening, a week after a 31-year-old woman and her two children, aged eight and three, were doused with a corrosive liquid.
Mr Savell told journalists that the massive manhunt for Ezedi is “an incredibly high priority attempted murder investigation”.
Turning to potential motives, he said: “They were in a relationship and that relationship had broken down.”
The latest confirmed sighting of Ezedi on CCTV is now on Vauxhall Bridge Road at around 11pm on the night of the attack, January 31.
Police on Wednesday released footage of him seen walking south on Westminster Bridge at 10.33pm.He has not used his cash card since taking the Tube from King’s Cross to Tower Hill earlier in the evening. Police recovered his phone from his car which he crashed attempting to drive away from the crime scene.
The woman who was attacked in Clapham remains sedated in hospital and is still too ill to speak to police.
She had agreed to meet Ezedi on the night of the attack, and she and her children were in a car with him when they were injured, police said.
Hundreds of calls have been received from the public with possible sightings of the suspect, who was initially using his bank card to travel around on the Tube network.
His route appears to broadly follow the River Thames, and investigators say they are keeping an open mind about whether he has jumped or been pushed into the water.
Commander Savell added: “We have worked with the National Crime Agency to commission medical experts to provide an assessment of Ezedi’s injuries.
“Those experts are clear Ezedi suffered a significant injury in the attack and is in increasing danger of an infection if he does not seek urgent medical attention. That will be extremely serious, if not life-threatening.
“To the best of our knowledge he has not been to hospital or sought professional medical help. At our request, all hospitals remain on high-alert. As time goes on, Ezedi’s health will continue to deteriorate.
”Ezedi – hand yourself into police, not only so we can speak to you, but so you can get the urgent medical treatment you obviously need.”
Darius Nasimi, from the charity the Afghanistan and Central Asian Association, appealed for anyone in the community who may be sheltering him to contact police.
He said: “This is a tragic incident for this mother and her two children. Violence against women and girls can not be tolerated.
“I would urge the Afghan community to help the police and provide any information at all about where Ezedi is, who is providing him with accommodation, who is harbouring him and helping him evade capture. You can also speak to our charity and we will help liaise with the police and pass information on if you feel you don’t want to speak to them directly.
“I make a direct appeal to him – hand yourself in. This has been going on for long enough. You need medical attention.
“What has happened in this case should have no reflection on the people of Afghanistan in the UK. This does not represent them.”
Counter terror police have been drafted in to aid the investigation and have been trawling through hundreds of hours of CCTV footage.
Updated timeline of Ezedi’s movements
The Met Police has released a timeline of Ezedi's last known locations
00:15 – Ezedi’s vehicle is seen in Newcastle
06:30 – His vehicle is then seen traveling into Tooting, London
16:30 – A further sighting of his vehicle is confirmed in Croydon
19:00 – He is then seen driving in Streatham
19:25 – Attack takes place in Lessar Avenue, SW4, before Ezedi makes off in his vehicle which crashes nearby. He leaves the car and runs off.
19:33 – Ezedi boards a train at Clapham South Tube Station.
19:59 – He is then seen leaving that train at King’s Cross Tube Station.
20:42 – He is then seen on CCTV leaving Tesco at 21 Caledonian Rd, London N1 9DX. He exits and turns right.
21:00 - Ezedi enters King’s Cross Tube Station and boards a Victoria Line tube southbound.
21:10 – He gets off at Victoria Tube Station and heads towards the district line.
21:16 – Ezedi boards an eastbound District Line train.
21:33 – Ezedi exits Tower Hill Tube Station.
21:47 - He is seen on Allhallows Lane, EC3. He travels through a passage to Cousin Lane. Then he turns right, walking towards Upper Thames Street.
21:51 - He then turns left onto Upper Thames Street
21:54 – He travels along Upper Thames Street
21:59 – Ezedi passes the City of London School on Pauls Walk EC4, heading towards Blackfriars Bridge. He passes the riverboat pier.
22:04 – He then passes the Unilever building and heads towards Victoria Embankment.
22:06 – Ezedi passes Carmelite House, junction with Carmelite Street towards Temple.22:28 - He passes Westminster Pier, goes up steps and continues towards Westminster.22:33 - Ezedi seen walking south on Westminster bridge (footage released today)
22:36 – He walks across Westminster Bridge, down steps on the London Eye side towards Lambeth
22:42 – Ezedi seen walking west on south Thames Path towards Lambeth Bridge
22:55 – He travels along the Albert Embankment approaching Vauxhall Bridge (image released today)
23:00 – Ezedi crosses Vauxhall Bridge towards stairs (Houses of Parliament side of the river)
23:03 - He crosses Vauxhall Bridge Road, into Grosvenor Road.
The suspect travelled from South Clapham tube, reappearing in a Tesco Express in Caledonian Road near King’s cross where he bought a bottle of water. The footage clearly shows he has serious injuries to the right side of his face. He then went into King’s Cross station and was picked up again on camera at Tower Hill.
He then headed towards the Thames with the last confirmed sighting when he was caught on camera crossing Vauxhall Bridge Road into Grosvenor Road at 11.03pm on Wednesday.
Ezedi is an Afghan refugee who was granted asylum despite being a convicted sex offender. He arrived in the UK on the back of a lorry in 2016.
He had been working at a pizza restaurant in Newcastle where he had set up home. He travelled from the city to London before launching the attack.
Witnesses described how the woman was shouting “My eyes, My eyes, I can’t see”, after he flung the corrosive substance at her.
Passers-by intervened and some also suffered burns.
He also threw the woman’s three year-old daughter to the ground like “a ragdoll”, eyewitnesses said.
Ezedi isn’t the father of the children in the attack, police have said.
Despite twice being refused asylum, a reference from a Baptist minister was crucial in persuading a 2020 immigration tribunal that he had changed religion, allowing him to stay in Britain.
But an Iranian asylum seeker, 33, who was Ezedi’s colleague at a pizza shop in Tyneside, said the fugitive came from a very strict and devout family.
He added: “The idea he is now a Christian is a joke but he loves money and he loves his passport so maybe he did it for that reason.”
A Baptist church in Tyne and Wear has confirmed it is aware of a “connection” it had with the suspect.
In a statement, the church in Jarrow said: “Grange Road Baptist Church brings together and welcomes people from all walks of life and from many different nations.
“We are aware of the connection Abdul Shakoor Ezedi has had with our church.
“As soon as we became aware of the current situation, we made contact with and are co-operating with the relevant authorities. We are praying for all those affected by the situation.”
Baptists Together, a movement of which the Grange Road church is a part, said it is “fully aware of the questions being asked of our churches surrounding Abdul Shokoor Ezedi and broader queries around supporting asylum seekers”.
They said: “Whenever anyone, asylum seeker or otherwise, explores Christian faith in a Baptist church, due care will be taken to ensure those wanting to profess Christian faith understand the deep commitment they are making and specifically the need to turn away from wrongdoing and seek to follow Christ in his ways of love.”
Ezedi is said to have been deeply affected by the murder of his sister shot dead by armed robbers trying to steal the money he had sent home to his family in Afghanistan from the UK.
His friend from the pizza shop said: “After what happened he was crying for three weeks and he said he wanted revenge. He was so upset and very, very angry.” However, detectives have established no link with his attack in Clapham.
A 22-year-old man arrested on suspicion of assisting an offender has been released on bail.