Employers will be told they will need to contribute to their furloughed workers' salaries from August - as the Government admitted the scheme cannot run "indefinitely".
He is facing calls - including from a cross-party group of 113 MPs - to extend the scheme supporting self-employed workers past its expiry on Sunday.
Critics say not extending the scheme risks leaving many "without work and without support".
Video report by ITV News Political Reporter Shehab Khan
The job retention scheme has so far covered the wages of 8.4 million staff in the UK unable to work during lockdown - at a cost of £15 billion.
Ministers have said they will extend the scheme covering 80 per cent of workers' salaries up to £2,500 per month until the end of October - though employers will be expected to make a contribution.
It's thought the Chancellor will ask employers to contribute around 20 per cent of wages, as well as National Insurance and pension contributions from August.
The move follows further easing of England's lockdown, announced by Prime Minister Boris Johnson at Thursday's daily briefing.
From Monday in England, friends and families able to meet in parks and gardens in socially distanced groups of six. Governments in the three other nations have also announced partial easing of their lockdown measures.
It comes as the UK's coronavirus death toll rose to 37,837 on Thursday, according to the latest figures from the Department of Health.
The Prime Minister said all five of his tests to move into the next phase had been met, allowing schools to begin reopening and greater contact to be permitted from Monday.
But the PM continues to be dogged by questions over his top aide Dominic Cummings after Durham police said he may have committed "a minor breach" of lockdown rules.
The easing of restrictions also means the gradual reopening of the economy, with outdoor retail and car showrooms opening in England from Monday ahead of a greater opening of non-essential shops.
Elsewhere, Environment Secretary George Eustice confirmed pubs with beer gardens are likely to be the first venues in the hospitality sector to reopen.
Mr Eustice said people cannot be furloughed "indefinitely" and ways need to be found to get people safely back to work.
On the issue of continued support for the self-employed, Mr Eustice told Sky News: "Well obviously it is nearly a month ago now that we said we wanted to reopen those bits of the economy that couldn’t work from home, so we’ve been encouraging the construction industry, for instance, to get back to work.
"A lot of those self-employed professions such as plumbers, electricians and so on, those people are able to return to work now, albeit observing social distancing, but we need to try to start to get bits of the economy back to work.
"Now I don’t know what Rishi Sunak, the Chancellor, will say later in terms of self-employed and the furlough scheme for them, but I think there is a general overarching message here that we've had a very generous furlough scheme in place to help people through these extraordinary times and to ensure that businesses’ overheads could be covered."
Shadow home secretary, Nick Thomas-Symonds, called on the Government to provide more financial support to the self-employed as well as those who will have to self-isolate under the new NHS Test and Trace system.
Mr Thomas-Symonds told BBC Radio 4's Today programme:
"There's a worry about the specific self-employed support scheme just abruptly coming to an end but there is also the broader question of the support that people receive when they do stay at home.
"And the availability of statutory sick pay, we’ve debated previously the level of statutory sick pay.
"The Prime Minister and the Chancellor have both made promises that people should not lose out for doing the right thing, they have to be held to that promise in the weeks and months ahead."
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