Theresa May has set a one-week deadline for cross-party Brexit talks to make progress or be brought “towards a conclusion”, with a crunch meeting likely to be scheduled with key players early next week.

Downing Street believes the middle of next week is the deadline by which the negotiations must show progress, if they are to have a hope of passing the withdrawal agreement and implementation bill (WAB), in time to prevent the European elections.

May’s spokesman said they had been “serious and constructive”, and added: “Further talks will now be scheduled in order to bring the process toward a conclusion.”

Downing Street now expects that cross-party talks with senior shadow ministers and officials to continue this week after participants on Monday emerged optimistic about a change in tone and a feeling there were grounds to continue discussions.

The talks are expected to culminate with a formal meeting of senior cabinet ministers next Tuesday or Wednesday, though sources on both sides said no new offer had been made yet by the government.

Some cabinet ministers, including the defence secretary, Gavin Williamson, have always been sceptical of the likelihood that the talks would yield a deal; while others, including Amber Rudd and Philip Hammond, have been more optimistic.

In the absence of an agreement with Labour under which both May and Jeremy Corbyn would whip their MPs to support a deal, the government hopes to secure opposition support for a process of votes in which MPs would consider a series of Brexit options.

At the cabinet meeting on Tuesday morning, David Lidington, updated the cabinet on the Brexit talks, while the Tory chief whip, Julian Smith, talked about the prospects for securing a majority for the WAB.

Lidington is regarded by Labour as one of the ministers most willing to compromise in order to secure support for a Brexit deal.

But the foreign secretary, Jeremy Hunt, underlined on Tuesday the extent to which the prime minister is boxed in, when he insisted that signing up to Labour’s central demand of a customs union could lose the government more Tory votes than it would gain from Labour.

Speaking on a tour of five African countries, Hunt said there was a “great sense of urgency” to get a deal passed, but that he “very much hoped” the talks would not end in a deal proposing a customs union.

“There is a risk you would lose more Conservatives than you gain Labour MPs,” the foreign secretary said. “If it was something different then the result could be different as well.”

The shadow chancellor, John McDonnell, who has been involved in the talks on the Labour side, criticised Hunt’s comments.

“This is hardly a helpful or constructive intervention whilst we are in the middle of cross-party talks to protect our economy and does not inspire confidence that if a deal is agreed it would be successfully entrenched and last any longer than the next Tory leadership election,” he said.

Hunt’s intervention came as May faced a growing revolt from grassroots activists in her own party.

Conservative association chairs have forced an emergency summit on the prime minister’s future, with 800 senior party officials including constituency chairs expected to hold a vote on whether she should resign.

The organiser of the summit said grassroots anger had been prompted by both the talks with Corbyn and the UK’s participation in the European parliamentary elections.

An extraordinary general meeting to discuss May’s future was triggered after more than 10% of chairs signed a petition calling for the summit, which is to be held in June.

The vote on her leadership would not be binding, but organisers have said they believe it would put significant pressure on May to resign and on cabinet ministers and MPs to act.

The Tory deputy chair, Andrew Sharpe, who chairs the National Conservative Convention, which organises grassroots members, has told May the meeting will be held, according to the Sun.

The petition was organised by the chair of the Bethnal Green and Bow Conservatives, Dinah Glover, who said she believed it was time for May to resign or for the cabinet to act to oust her.

MPs cannot hold a vote of confidence in the prime minister until December, after May won a year’s grace when she defeated a no-confidence challenge before Christmas last year. She has pledged to step down when the first phase of Brexit is delivered.

The motion drafted by Glover says the party “no longer feel that Mrs May is the right person to continue as prime minister to lead us forward in the negotiations”.

It adds: “We therefore with great reluctance ask that she considers her position and resigns, to allow the Conservative party to choose another leader, and the country to move forward and negotiate our exit from the EU.”

When the petition was launched, Glover said the triggers for most constituency chairs to sign the petition had been the “humiliation” of holding European elections and May’s decision to enter talks with the Labour leader to find a Brexit compromise.

• This article was amended on 1 May 2019 because an earlier version said 800 constituency chairs were expected to vote on Theresa May’s future. The 800 include senior Conservative officials and the heads of the party’s constituency associations.