Boris Johnson has promised a renewed effort to secure a deal with the EU before the Brexit deadline.
The UK’s negotiators will now meet EU counterparts twice a week in the run-up to a crucial summit on 17 October.
It follows a backlash from MPs and opponents of a no-deal Brexit against the prime minister’s decision to suspend Parliament next month.
Former Conservative PM Sir John Major has announced he is backing moves for a judicial review of Mr Johnson’s plan.
And another senior Tory, Sir Oliver Letwin, said MPs still had time to act to prevent the UK leaving the EU without a deal.
As things stand, the UK is due to leave the EU on 31 October with or without a “divorce” agreement.
The EU said it expected the UK to come up with “concrete proposals” in the near future.
A European Commission spokesperson said its “doors remain open” and insisted it had “demonstrated our willingness to work 24/7 throughout this long process”.
Sir John Major, who has previously threatened legal action against the prorogation – suspension – of Parliament, announced on Friday that he was backing campaigner Gina Miller’s application for a judicial review of Mr Johnson’s decision.
This would “avoid duplication of effort” and was necessary because of the “imminence of the prorogation”, he added.
In a separate development, a Scottish judge has refused to order a temporary halt to the prime minister’s plan.
The previous government, under Theresa May, agreed a deal with the EU but it was rejected by MPs three times.
Mr Johnson says, while he would prefer to reach a new deal, he is willing to leave without one – and maintains the UK will leave by the October deadline “no ifs, no buts”.
A number of MPs who oppose a no-deal Brexit are planning to take action in Parliament next week to force Mr Johnson to ask the EU for an extension to the deadline if a deal is not agreed in time.
The PM’s decision to suspend Parliament for five weeks in September and October has been criticised as a tactic to prevent such action.
But No 10 insists it is normal process to prompt a Queen’s Speech and allow the new government to put forward its “very exciting agenda”.
Mr Johnson said he had been “encouraged” by the response from EU leaders but it was “now time for both sides to step up the tempo”.
He also said he had seen “a willingness to talk about alternatives to the anti-democratic backstop”.
The backstop – a controversial back-up plan to prevent a hard border in Ireland if no post-Brexit trade deal is agreed – is likely to be at the forefront of the twice-weekly discussions between Brussels and the prime minister’s lead negotiator, David Frost.
But Ireland’s Foreign Affairs Minister, Simon Coveney, said it was up to the UK to “propose alternatives that can solve those problems”.
What are MPs’ plans to stop no deal?
MPs are remaining tight-lipped about their precise plan to stop a no deal.
But Labour leader Jeremy Corbyn said the party would attempt to bring legislation as soon as Parliament returns, and that a no-confidence vote to bring down the government remains an option.
BBC Newsnight’s political editor Nick Watt said a cross-party group of MPs was planning a “surgical strike”, involving an emergency debate on Tuesday and a vote to take control of the Commons agenda the following day.
That could prevent Chancellor Sajid Javid from carrying out his spending review, scheduled for Wednesday.
It is also thought some MPs are exploring ways of ensuring Parliament can meet on the Friday, Saturday and Sunday before the planned suspension.
Sir Oliver said, despite Mr Johnson’s decision to suspend Parliament, he still believed MPs had time to prevent a no-deal scenario.
He told BBC Radio 4’s Today programme: “I hope that Parliament will take a series of actions in a proper, orderly way that means, by the end of the week, Boris Johnson knows that as prime minister he has the backing of many of us to get a deal.
“But if he doesn’t get a deal, he is going to have to seek an extension.”
Shadow attorney general Shami Chakrabarti also told the programme she believed there was now enough support among MPs to stop no deal, saying the numbers had been “strengthened” by rebel Conservatives following the prorogation announcement.
But their plan also faces possible challenges in the House of Lords, where Brexiteers could table multiple amendments as a delaying tactic until the shutdown begins.
However, Lord Lisvane – the former clerk of the House of Commons – told Today he believed the Lords “would be very cautious about seeking to kill a bill which had actually been passed by an elected house”.
Can the rebel alliance stop no-deal Brexit?
Within days we will know if the MPs who are implacably opposed to leaving the EU without a deal can really stop that happening.
With lots of former ministers on the backbenches, the group which is openly fighting against the possibility of leaving the EU without a deal has a different complexion.
But given Boris Johnson’s main reason for success in the Tory leadership election was to leave the EU whatever it takes at the end of October, it is a pretty major goal for a group of backbenchers and opposition politicians.
What is the reaction to the suspension of Parliament?
There has been outrage from all sides of the House about prorogation, with Labour, the SNP, the Lib Dems, Plaid Cymru, the Independent Group for Change and the Green Party releasing a joint statement to condemn it as “undemocratic”.
Some Tories have also condemned the decision, with government whip Lord Young resigning in protest.
But Foreign Secretary Dominic Raab said the idea it was some kind of constitutional outrage was “nonsense”, and the public wanted to see the government getting on with things other than Brexit.
The planned suspension triggered protests across the country, with further demonstrations expected this weekend.
Meanwhile, an e-petition on Parliament’s website against the suspension reached more than 1.5 million signatures.