Mr Johnson will update the Cabinet on plans to lift the lockdown on Monday, but it is unlikely that any easing will come into effect before the start of next week.
Northern Ireland already allows groups of up to six people from different households to meet outdoors, so long as they maintain social distancing.
The Welsh government still prevents members of different households from meeting outdoors, although the health minister is reviewing the policy and a review is promised on Thursday.
Meanwhile Nicola Sturgeon has promised to allow outside meetings between a maximum of two households from that date.
The Scientific Advisory Group for Emergencies is understood to have advised ministers that Covid-19 infection is far less likely outdoors.
The Government’s roadmap document states: “The intention of this change would be to allow those who are isolated some more social contact, and to reduce the most harmful effects of the current social restrictions, while continuing to limit the risk of chains of transmission.”
“This could be based on the New Zealand model of household ‘bubbles’ where a single ‘bubble’ is the people you live with.”
Step 2 of the roadmap also includes the opening of non-essential retail outlets, subject to the size of the establishment and the ability to enforce social distancing.
Outdoor markets and car show rooms are two examples, a source told The Daily Telegraph.
However, any reopenings as part of Step 2 are not thought to include hospitality venues such as pubs and restaurants.
Some of these venues could re-open from July 4, subject to incidence of the virus decreasing sufficiently, according to the roadmap document.
“We must keep that R down below one, and that means we must all remember the basics: wash our hands, keep social distance, isolate,” said Mr Johnson on Sunday.
“We are beating this thing. But we will beat it all the faster if we stare that control the virus and save lives.”
Aggressive monitoring of new infections is a critical component of the route out of lockdown.
Matt Hancock, the health secretary, has recruited 25,000 contact tracers to track down people who may have come into contact with newly-infected patients.
However, some experts have said that not enough volunteers will be deployed in the community, rather than call centres.