Mr Jenrick told Sky News that "like many people, I want to get away from these restrictions as quickly as I possibly can”.
“We don't want them to stay in place for a day longer than is necessary.
"I think we are going to now move into a period where there won't be legal restrictions, the state won't be telling you what to do, but you will want to exercise a degree of personal responsibility and judgment - different people will come to different conclusions on things like masks.”
Mr Jenrick said Prime Minister Boris Johnson would set out more detail on the lifting of restrictions in coming days.
He also said he would not personally wear a mask after July 19, if permitted to do so. "I will. I don't particularly want to wear a mask. I don't think a lot of people enjoy doing it."
Separately, the newly appointed Health Secretary Sajid Javid said there were “compelling” health reasons to ease lockdown measures.
Mr Javid wrote in the Mail on Sunday: “The economic arguments for opening up are well known, but for me, the health arguments are equally compelling.
“The pandemic has hit some groups disproportionately hard. Rules that we have had to put in place have caused a shocking rise in domestic violence and a terrible impact on so many people’s mental health.”
Meanwhile, Downing Street on Saturday confirmed proposals to scrap quarantine requirements for those who have received two doses of a coronavirus vaccine were under “consideration”.
A No 10 source said it was looking at whether to drop all legal self-isolation measures for fully vaccinated people who come into contact with someone who is infected as a possibility for life after Stage 4 of the road map.
The hospitality industry, however, said such a move would not lift pressures it is currently under as a result of the NHS Test and Trace app, which bosses said was “casting the net quite wide” in terms of who it pings as a close contact of a positive case, leading to mass shutdowns of venues.
Emma McClarkin, chief executive of the British Beer and Pub Association, said pubs were “closing or greatly reducing their opening hours due to staff shortages caused by app pings”, despite staff testing negative for coronavirus using quick-result tests.
But reports have suggested the call for action from the trade has been heard by ministers, with The Sunday Times stating that the need to scan a QR code before entering bars, restaurants and other venues, such as museums, will become a thing of the past after July 19 – a change that would mean less chance of punters and staff being told to self-isolate by Test and Trace.
The possibility of a wholesale easing of restrictions will come as a blow to senior doctors, who had petitioned for some measures to remain in place to arrest the “alarming” rise in Covid-19 cases in England.
The Government confirmed that, as of 9am on Saturday, there had been a further 24,885 lab-confirmed Covid-19 cases in the UK, while a further 18 people had died within 28 days of testing positive.
It comes as the latest figures from the Office for National Statistics (ONS) show that about one in 260 people in private households in England had Covid in the week to June 26 – up from one in 440 in the previous week and the highest level since the week to February 27.
The British Medical Association (BMA) had said that keeping some protective measures in place was “crucial” to stop spiralling case numbers having a “devastating impact” on people’s health, the NHS, the economy and education.
Asked about the reports of lockdown easing, a Downing Street spokeswoman said: “No decisions have been made but more details will be set out soon.”