Mr Sunak also announced more details of how the furlough scheme for employees will be paid for in the coming months.
Employers will also be expected to pay national insurance contributions, which they are now exempt from, and employer pension contributions.
The Indian-origin finance minister on Friday said the furlough, or forced leave, scheme, under which the United Kingdom taxpayer covered 80 per cent of the wages of almost a million jobs, will begin to wind down with employers asked to contribute a "modest" proportion of those costs from August before it ends in October.
At the moment the Government is paying the wages of 8.4million people, but the PM has said he wants people to start to return to work if they can.
As part of the changes, in June and July, the scheme will continue as before with the government covering 80 per cent of wages up to GBP 2,500 with no employer contribution.
Mr Sunak said: "Our top priority has always been to support people, protect jobs and businesses through this crisis".
The vehicle retailing businesses has clearly changed and is likely to change forever.
Employers groups have warned that a dramatic change to the scheme at the end of July will lead to tens of thousands of redundancies.
Employers should pay the staff normally for the hours they are working - and then claim the rest from the government.
Those who qualified for the Treasury's self-employment income support scheme would be able to apply for another payment, calculated as 70% of their monthly profits, up to a ceiling of £6,570.
At the No 10 briefing, Mr Sunak also set out how employers will have to start sharing the cost of the scheme.
"The economic recovery is still in its infancy and it will be some time before businesses are back to their pre-pandemic trading level". He extended another multi-billion-pound income scheme for self-employed workers as well. Officials are expected to present revised figures next week.
Millions of workers have been thrown a financial lifeline after the Government extended its support scheme for the self-employed by three months.
A scheme to assist the self-employed has also been continued, with beneficiaries able to receive another three-month grant in August, based on 70 per cent of their previous earnings. Applicants will need to confirm their work has been affected by the virus, but they would not need to have taken the first grant to be eligible for the second.
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