The extremist group claimed the 28-year-old convicted terrorist was one of its fighters but did not provide any evidence.
No one else is being sought over Friday’s attack, which claimed the lives of two people and left three others injured.
Video footage from 2008 shows Khan, who would have been about 17 at the time, telling reporters “I ain’t no terrorist” as anti-terror police raided his home.
He told the BBC: “I’ve been born and bred in England, in Stoke-On-Trent, in Cobridge, and all the community knows me and they will know, if you ask them, they will know like these labels what they’re putting on us, like terrorist, this, that, they will know I ain’t no terrorist.”
Shortly before 2pm on Friday, Khan, armed with two knives and wearing a fake suicide vest, stabbed 25-year-old Jack Merritt and a woman to death and left three other people injured after going on a knife rampage in the capital.
The convicted terrorist was on licence and wearing an electronic monitoring tag when he attended a conference on prisoner rehabilitation hosted by Cambridge University scheme Learning Together at Fishmongers’ Hall near London Bridge.
Khan, who was living in Stafford, was given permission to travel into the heart of London by police and the probation service. He had also been allowed to travel to Whitehall earlier in the year.
He was tackled to the ground by members of the public, including ex-offenders from the conference, before he was shot dead by police in front of shocked bystanders on London Bridge.
Tales of acts of bravery quickly emerged as footage was shared online showing Khan being taken to the ground as one man sprayed him with a fire extinguisher and another, reportedly a Polish man who worked at the Hall, lunged towards him with a narwhal tusk believed to have been taken from a wall inside the building.
Khan was part of an al-Qaieda-inspired terror group – linked to radical preacher Anjem Choudary – that plotted to bomb the London Stock Exchange and build a terrorist training camp on land in Pakistan-controlled Kashmir owned by his family.
A list of other potential targets included the names and addresses of the Dean of St Paul’s Cathedral in London, then London mayor Boris Johnson, two rabbis, and the American Embassy in London.
He was released from prison in December 2018 less than seven years into a 16-year prison sentence.
Friday’s attack has prompted the Ministry of Justice to review the licence conditions of every convicted terrorist released from prison, which is understood to be about 70 people.
Mr Merritt’s father David called his son a “beautiful spirit who always took the side of the underdog”, adding: “He was an exceptional young man, and I’m only finding out the half of it now he’s gone.”
Additional reporting by Press Association.