Easing the restriction is seen as vital if businesses such a restaurants and pubs are to be able to re-open sustainably.
The Mail on Sunday reported the review would effectively take control of social-distancing guidelines out of the hands of the Government's scientific advisers, who have been deeply reluctant to countenance relaxation.
The move comes as thousands of non-essential shops in England are set to re-open on Monday for the first time since the coronavirus lockdown was imposed in March.
With many people thought to be nervous about going out again after nearly three months in lockdown, Business Secretary Alok Sharma sought to reassure the public that measures had been put in place to ensure their safety.
Writing in the Sunday Express, he said: "We need to get Britain's economy firing again, while at the same time making sure we keep people safe and avoid a second peak of the disease."
And Chancellor Rishi Sunak told the Sun on Sunday: "I am very conscious that there will be anxiety. For some time, many people have not been inside a shop and, in a way, we all have to relearn the behaviours we took for granted.
"We've been living with anxiety now for 12 weeks but the good news is that we've made enormous progress.
"Bit by bit, that confidence will come back and the anxiety will reduce. But it's not going to happen overnight."
There have been warnings that any maintaining of the two-metre rule, along with a closing of the furlough scheme, could be a "horrendous" situation for the hospitality sector.
Richard Caring, chairman of Caprice Holdings which runs the Ivy, accused the Government of "killing the country" in the Mail on Sunday.
He said: "There are estimates saying we could have up to five million unemployed. It's not going to be five million - it's going to be more. I don't think we've seen anything yet.
"The Government is actually killing the country right now and the hospitality industry is in the front line of the disaster."
Meanwhile, the Government is set to mount a fresh push to have more primary school children back into the classroom ahead of the summer break.
With most children in England set to remain at home until September, ministers have been accused of putting retail before pupils' education.
A No 10 source said Mr Johnson was "acutely aware" of the impact the extended closure was having on pupils and was working with Education Secretary Gavin Williamson on a major "catch-up" plan.
Currently primary schools in England - which closed following the coronavirus lockdown in March - are opening to pupils in Reception, Year 1 and Year 6.
However, ministers will this week reaffirm schools can take children from other year groups provided they have the capacity to do so safely.
It means limiting class sizes to just 15 while ensuring protective measures are in place to prevent the spread of the coronavirus.
The details came as the Children's Commissioner for England Anne Longfield issued a fresh warning that the failure to re-open schools risked undermining children's basic right to an education.
"It has taken 200 years of campaigning to get children into the classroom, ensuring that education was a basic right for all children," she told the Observer.
"We seem for the first time to be prepared to let that start go into reverse. And I think that is a very, very dangerous place to be.
"We heard from the Prime Minister back in April that education was one of the top three priorities for easing lockdown, but it seems to have been given up on quite easily."
With most children not now due to return until September, it will have been nearly six months since they have been in a classroom by the time they get back.
The Prime Minister was said to be particularly concerned about the impact on disadvantaged children who lack the same support at home and access to remote learning as others.
A No 10 source said: "The PM is acutely aware that school closures will have a disproportionate impact on all children, and particularly the most disadvantaged and vulnerable children.
"He appreciates the consequences of months out of school, and this package will be focused on providing extended support for children.
"The PM is so grateful for the hard work of teachers, parents and schools to keep educating children throughout this difficult period."
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