Theresa May's future as prime minister is hanging in the balance as Brexit-supporting members of the 1922 Committee believe they are close to securing enough support for another attempt to oust her.
May will meet Graham Brady, the chairman of an influential committee of backbench Tory MPs, as there have been calls for a firm resignation date.
It followed a request from the 1922 Committee for "clarity" on the issue. The National Conservative Convention's vote on 15 June is non-binding, though, so the PM could ignore it.
For now, the United Kingdom government aims to exit the European Union latest by August 1.
His party is locked in talks with the government in a bid to reach an agreement on how Britain will leave the European Union that will get through Parliament.
"The time has come for her to resign", said lawmaker Bill Cash.
European elections on 23 May will go ahead, almost three years after the Brexit referendum, the government has confirmed.
However, while Labour will want to string out the talks, they will not want to join what Mr Farage has already branded "a coalition against the people" to help Mrs May's deal through.More news: The Island | #AsiaNewsNetwork: More attacks feared in Colombo, police say
But Chancellor Philip Hammond defended the cross-party talks, suggesting the government had no other option.
The report cited unnamed government sources, who said the referendum plan would only become relevant if talks with Labour failed and a majority in parliament supported holding another public vote.
Media captionConfused by Brexit jargon?
After Prime Minister Theresa May's deal was rejected three times and she was forced to delay Brexit, the government has spent more than four weeks in talks with Labour - negotiations that have done little to soften positions in either party.
Pressure is building on Mrs May following last week's local election drubbing, in which the Conservatives lost 1,334 councillors in England.
"It's aimed at Conservative supporters who think the prime minister is doing her best to get this over..."
May is preparing to make new proposals of a temporary customs union at least until the next election in 2022.