The final decision on whether the change can be brought in from June 21 will depend on the data, the Prime Minister added.
Mr Johnson said he feels like the next stage of reopening on May 17 - which covers indoor hospitality, entertainment and possibly foreign travel - "is going to be good".
One industry chief has said a return to unrestricted trading for hospitality from June 21 is "critical" and will mean firms can "come off life support".
The comments from Kate Nicholls chief executive of trade body UKHospitality, came as The Times reported social distancing rules will be lifted to allow pubs, restaurants and theatres to open to full capacity for the first time in more than a year.
One-way systems, screens and mask-wearing while moving around might remain for hospitality venues but customer numbers will no longer be limited, the newspaper said.
Audiences in theatres and cinemas will have to wear face coverings during performances, while there will be strict guidance on ventilation and staggered entry, The Times reported.
Speaking during a campaign visit to Hartlepool, Mr Johnson told reporters: "As things stand, and the way things are going, with the vaccine rollout going the way that it is - we have done 50 million jabs as I speak to you today, quarter of the adult population, one in four have had two jabs.
"You are seeing the results of that really starting to show up in the epidemiology.
"I think that we will be able to go ahead, feels like May 17 is going to be good.
"But it also looks to me as though June 21 we'll be able to say social distancing as we currently have to do it, the one-metre plus, I think we have got a good chance of being able to dispense with the one-metre plus from June 21.
"That is still dependent on the data, we can't say it categorically yet, we have got to look at the epidemiology as we progress, we have got to look at where we get to with the disease. But that's what it feels like to me right now."
On Sunday, Foreign Secretary Dominic Raab said "some safeguards" may stay in place when legal restrictions end, such as continued use of masks and physical distancing.
Last month, Government scientific advisers said the public should be able to remove face masks over the summer as vaccines do the heavy lifting in controlling Covid-19 - but they cautioned that masks and possibly other measures may be needed next autumn and winter if cases surge.
Responding to the report in The Times, Ms Nicholls said: "These reports are very welcome if true.
"However, we must wait to see the full detail of plans as any restrictions in venues will continue to impact revenue and business viability.
"A return to unrestricted trading on June 21 is critical and will mean hospitality businesses (can) come off life support and be viable for the first time in almost 16 months.
"We urge the Government to confirm reopening dates and these plans at the earliest opportunity, which will boost confidence and allow companies to step up planning and bring staff back."
A spokesman for the UK Cinema Association indicated that the organisation hopes face coverings will not be a continued requirement.
He said: "We strongly believe that our exemplary record on safety - with not a single case of Covid traced back to a UK venue - and our ability to manage the movement of cinema-goers in modern, highly ventilated indoor environments offer ample evidence that any relaxation from June 21 can be undertaken safely without the need for further ongoing restrictions, including any requirement for face coverings."
A Cabinet Office spokesman pointed back to the wording of the road map out of lockdown, which states that the Government "will complete a review of social distancing measures and other long-term measures that have been put in place to limit transmission".
The review's findings "will help inform decisions on the timing and circumstances under which rules on one-metre-plus, face masks and other measures may be lifted".
It comes as the Government announced the limit on the number of mourners who can attend funerals is to be lifted in England.
The Ministry of Housing, Communities and Local Government (MHCLG) said the legal restriction of a maximum of 30 mourners will be removed as part of the next stage of lockdown easing, expected on May 17.
The capacity will be determined by how many people venues, such as places of worship or funeral homes, can safely accommodate while maintaining social distancing, the department added.
The ban on foreign holidays is expected to be lifted for people in England from May 17 as part of the next easing of coronavirus restrictions.
But Mr Johnson has cautioned that while there will be "some openings up" from that date, the approach must be "sensible" to avoid an "influx of disease" when international travel resumes.