Boris Johnson is announcing his second major review of coronavirus lockdown rules, 66 days after he introduced them in late March.
But growing numbers of Tory MPs are claiming his review has been undermined by his refusal to sack his aide Dominic Cummings for breaking the rules.
The number of Conservative MPs demanding Mr Cummings' dismissal is now approaching 50, with many more publicly criticising his actions.
The former chancellor Sajid Javid said Mr Cummings's trip to Durham was neither "necessary or justified" and he should apologise.
And the Paymaster General Penny Mordaunt became the first minister to speak out, claiming the row was undermining public health messages.
The prime minister is expected to give the all-clear for schools in England to begin reopening next Monday and non-essential shops a fortnight later, on 15 June.
But during a 99-minute interrogation by the Liaison Committee of senior MPs, dominated by clashes over Mr Cummings, he hinted there could also be good news on pubs and restaurants.
He said the government was "trying to go as fast as we can" to reopen the hospitality industry, which in the previous review on 7 May he said he hoped would happen on 3 July.
"It is very difficult to bring forward hospitality measures in a way that involves social distancing," he told the Liaison Committee during questions on reopening the economy.
"But I'm much more optimistic about that than I was. We may be able to do things faster than I had previously thought."
The PM also said he hoped the two-metre social distancing rule could soon be reduced to enable businesses such as pubs and restaurants to reopen faster.
He said he had asked the government's scientific advisers to look again at the guidance and was optimistic "we will be able to reduce that distance" as the coronavirus lockdown is lifted.
Downing Street has revealed that the next steps in easing the lockdown in England are likely to be unveiled at the daily news conference, likely to be fronted by the prime minister.
But it was emphasised that the final decision before the latest three-week legal deadline would not be made until after ministers considered evidence from the SAGE group of scientific experts.
Key to ensuring the restrictions can be safely relaxed is keeping down the so-called R rate, the average number of people to whom an infected individual transmits the disease.
Mr Johnson told the Liaison Committee "not everybody will go back" on day one of the phased reopening of schools next week, but insisted it was safe provided everybody remembered guidelines on social distancing and hand washing.
It has also been reported that under new plans to "increase social contact" two households will be able to meet outdoors from next month.
That would mean it would be possible to enjoy a picnic in a public space with another family or visit friends in a garden.
But the prime minister, who repeatedly told the Liaison Committee it was "time to move on" from the Cummings row, will be worried that criticism of his close aide is continuing to grow and now being joined by senior figures.
Calling on Mr Cummings to issue an apology, Mr Javid said: "I do not believe Mr Cummings' journey to County Durham to isolate on his family's estate was necessary or justified.
"I remain unconvinced his visit to Barnard Castle could be considered reasonable.
"I was also deeply concerned by his decision to return to Downing Street directly after coming into contact with a family member who was ill, potentially with coronavirus."
Ms Mordaunt, sacked from the cabinet by Mr Johnson when he became PM but brought back into government after the election, said there were inconsistencies in Mr Cummings's account of events and his reasons.
"There is no doubt he took risks - refuelling at a petrol station is a risk to oneself and to others, which presumably he did," she said.
"What is clear is that the scenes of the last few days will have undermined key public health messages. I deeply regret this and am very sorry for it."
Earlier, during the Liaison Committee, Mr Johnson rejected calls from MPs for an inquiry by the Cabinet Secretary, Sir Mark Sedwill.
"Quite frankly I'm not certain that an inquiry is a good use of official time," the prime minister said, claiming Sir Mark was busy dealing with the coronavirus pandemic.
Next week from Monday to Thursday, Dermot Murnaghan will be hosting After the Pandemic: Our New World - a series of special live programmes about what our world will be like once the pandemic is over.
We'll be joined by some of the biggest names from the worlds of culture, politics, economics, science and technology. And you can take part too. If you'd like to be in our virtual audience - from your own home - and put questions to the experts, email email@example.com