Senior Conservatives hope Brexit compromise talks will conclude with enough time for the UK to avoid fighting European Parliament elections.
Cabinet Office minister David Lidington and Iain Duncan Smith both indicated they wanted negotiations to wrap up before 22 May.
Britain has been granted a new Brexit delay until 31 October by the EU, with the option to pull out of the bloc earlier if parliament passes a divorce deal.
MPs have three times voted down the agreement, leading to missed Brexit deadlines of 29 March and 12 April.
But on Sunday, the two senior figures from the Conservative front and back benches both voiced hopes talks with Labour to break the parliamentary deadlock would wrap up soon.
Mr Lidington, the de facto deputy prime minister, said that while the cross-party talks would continue during MPs’ Easter break on specific policy subject areas, he would not let the issue “drag out”.
He announced that when parliament returns on 23 April then we will “take stock of where we are” – and if the talks fail then indicative votes will be held.
Speaking to the BBC’s Andrew Marr Show, Mr Lidington denied there was a “particular date ringed in the calendar” as a deadline but insisted the public “want politicians to get on and deal with this” and the process would last weeks not months.
It comes after Jeremy Corbyn was urged to back a second referendum or risk a wipeout at the next election.
For the Brexiteers, Mr Duncan Smith gave a stronger warning, saying delaying Brexit meant “political death” for the Conservative Party.
Asked on Sky News’ Sophy Ridge on Sunday show if there could be a grassroots revolt at the prospect of campaigning in the European election, he said: “There already is, to be quite frank.
“People don’t want to fight the Euro elections, they want to get on with the May [local] elections and they want to do it in a sense that they can say to the public: ‘We promised you we would leave, we are and we have left so we will leave before the Euro elections take place.'”
He called the prospect of holding them a “disaster” and called on the prime minister to rule out participating in them, meaning that under the EU’s latest extension offer the UK would to be on track to leave without a deal on 1 June.
Mr Duncan Smith said: “If we make that clear today and say we are going to leave, deal or no deal, before the Euro elections then I think the public would start to snap back and say: ‘Okay, these people mean business.’
“And when we do it, that’s the moment that we’ll end all of the nonsense from these other peripheral parties like UKIP and the Brexit Party and the Tiggers (The Independent Group).”
Mr Duncan Smith said after the first time Brexit was delayed, “that’s when this has all gone wrong”.
“They were expecting us to go and when we didn’t go it looked like a complete breach with the pledge that we had made and that’s a disaster for a political party,” he said.
“There is no question that polling at the moment, I think it’s very voluble and very flexible but it’s where it is now and I suspect it’s about right.”
It comes after polling analysis in the Sunday Telegraph said 59 Conservative MPs would lose their seat in a snap general election.
Those at “high risk” of being ejected from the Commons included Mr Duncan Smith and Work and Pensions Secretary Amber Rudd.