Tory leadership contender Matt Hancock has promised to hold a vote on his Brexit deal before the end of July.
The health secretary said on Tuesday he would ask MPs to back his plan “immediately” after becoming prime minister.
Theresa May’s successor is due to be announced on Monday July 22 and be in place before the Commons rises for its summer recess.
Speaking to BBC Radio 4′s Today programme, Hancock said a positive vote would “show the European Union that this plan is deliverable through the House of Commons”
Hancock has claimed he will be able to persuade the EU to put a time limit on the Northern Ireland backstop - despite Brussels having rejected this idea in the past.
“They nearly proposed a time limit on the backstop before,” Hancock said, “but they didn’t think that the prime minister, Theresa May, would be able to get it through the House of Commons.”
In a message that will be interpreted as a challenge to frontrunner Boris Johnson, Hancock also said all the challengers for No.10 should “be open to scrutiny”.
Johnson has studiously avoided TV and radio interviews since announcing he would run for the leadership.
“I think everybody should participate in the proposed TV debates and I think we have got to ask the question ‘Why not?’” Hancock said.
He added the candidates should “come and be scrutinised because that is the best way to ensure that we get the best next prime minister”.
Ten candidates will go into the first round of voting on Thursday. Andrea Leadsom, Mark Harper and Rory Stewart will all formally launch their campaigns on Tuesday.
Michael Gove, Jeremy Hunt, Dominic Raab, Esther McVey and Hancock launched their campaigns on Monday. Johnson, Sajid Javid are due to follow this week.