Theresa May has vowed to Tory grassroots activists that she will not allow the referendum vote for Britain to leave the EU to be frustrated.

The prime minister is flying to Egypt for an EU-League of Arab States summit where she is expected to hold talks with key EU figures as she battles to break the deadlock in the Brexit talks.

Downing Street has played down hopes of a breakthrough during the course of the two-day gathering in the Red Sea resort, despite the presence of major players including the European Council president, Donald Tusk.

The prime minister is pressing for changes to the Northern Ireland backstop which she hopes will finally convince MPs to back her Withdrawal Agreement following last month’s crushing Commons defeat.

Ahead of her departure, No 10 released details of her speech to a closed meeting of the National Conservative Convention (NCC) in Oxford on Saturday, when she told supporters the government’s focus on delivering Brexit must be “absolute”.

Her comments came after three pro-EU cabinet ministers signalled they could back moves in parliament to delay Britain’s withdrawal to prevent a “disastrous” no-deal break.

The intervention by Amber Rudd, Greg Clark and David Gauke led to calls for their resignations by furious Tory Brexiters - comments said to have been echoed in private by some cabinet ministers.

The Northern Ireland minister, John Penrose, warned taking no-deal off the table could undermine May’s efforts to secure concessions on the backstop.

“It could torpedo Brexit completely, leaving us in a ‘Hotel California’ Brexit, where we’d checked out but could never leave,” he said in an article for the Sunday Telegraph.

“We’d have built an enormous elephant trap for ourselves, and there’d be no way to climb out.”

MPs are preparing for a potentially crucial series of votes on Wednesday which could see parliament seize control of the Brexit process if May cannot secure an agreement with Brussels by mid-March.

However, the Sunday Times reports the votes could now be shelved in the wake of the latest cabinet divisions.

In a joint article, Rudd, Clark and Gauke said it was clear a majority of MPs would back an extension to the Article 50 withdrawal process rather than see no deal.

However, in her address to the NCC, May insisted it is vital the government maintains its focus as the negotiations reach their final stages.

“Our focus to deliver Brexit must be absolute. We must not, and I will not, frustrate what was the largest democratic exercise in this country’s history,” she said.

“In the very final stages of this process, the worst thing we could do is lose our focus.”

With party loyalties strained to breaking point following the resignations of three Tory MPs to join a group of Labour defectors in the new Independent Group, May warned against more internal blood-letting.

Anna Soubry, Heidi Allen and Sarah Wollaston all complained they had been targeted in their constituencies because of their pro-Remain views, in some cases by former Ukip members who had switched to the Tories.

However, May said: “We are not a party of purges and retribution. We called a referendum and let people express their views – so we should not be seeking to deselect any of our MPs because of their views on Brexit.

“Our party is rightly a broad church – on that and other issues. And we will only save our country from the threat of Jeremy Corbyn if we remain one.”

On Wednesday, the Commons is expected to consider an amendment tabled by Labour MP Yvette Cooper and Conservative former minister Sir Oliver Letwin enabling the House to extend the Article 50 withdrawal process if there is no deal by mid-March.

A similar amendment was defeated by MPs last month, but there is speculation that enough Tory rebels, alarmed that there is still no deal in place, could be prepared to back it this time round for it to pass.

Downing Street has said if there is no deal by Tuesday, the prime minister will at that point make another statement to the House and table an amendable motion to be debated and voted on the following day.

However, it is unclear whether that will be enough to stave off a revolt by MPs alarmed at the prospect of no deal – including potential ministerial resignations.

• This article was amended on 25 February 2019. John Penrose is a Northern Ireland minister, not “Northern minister” as an earlier version said.