Conservative leadership hopeful Dominic Raab has warned the party will be "toast" unless it delivers Brexit by the 31 October deadline.
Speaking on Sky News' Sophy Ridge on Sunday programme, the former Brexit secretary also defended his refusal to rule out suspending parliament in order to ensure the UK's withdrawal, arguing that the EU needed to know that Britain was "serious".
However, in an interview with Ridge, Work and Pensions Secretary Amber Rudd branded the suggestion of shutting down the Commons as "ridiculous" and said promising Brexit by the end of October was misleading as the Commons Speaker was an "activist" in blocking no-deal.
Labour chairman of the Brexit select committee Hilary Benn told Sky News the prorogation of parliament to push through Brexit would be "scandalous".
In what will be seen as a thinly veiled attack on the race favourite Boris Johnson, Ms Rudd also criticised the "do or die approach" being taken by some of the leadership candidates towards Brexit.
Ms Rudd also said she was "still thinking very carefully about any lifts home from Boris".
This was a reference to a barbed remark Ms Rudd, a Remainer, made during the 2016 Brexit referendum campaign when she described Mr Johnson as the "life and soul of the party" but "not the man you want driving you home at the end of the evening".
Mr Johnson has also faced criticism over his reluctance to face media scrutiny, with concerns among his campaign team that he could throw away his clear lead.
He has made clear he will not take part in the first television debate being staged on Sunday by Channel 4.
However, he has indicated he will be at a BBC debate on Tuesday when the field of candidates will have been further whittled down in the second round of voting.
Sky News is planning to host a live leadership debate between the two final contenders.
As the second week of the campaign got under way, Mr Raab told Ridge: "The Tory Party will be toast unless we are out by the end of October. The Conservatives cannot win an election unless we have delivered Brexit."
Although it was very unlikely that parliament would be suspended over Brexit, he refused to rule it out.
Mr Raab said: "What is really scandalous here is the way that people are trying to sabotage the will of the people and break their promises left, right and centre to get us out of the EU.
"The big mistake we made in these negotiations was taking no-deal off the table. When we start ruling things out we only weaken our chances of getting a deal.
"All those candidates that are going weak at the knees and saying 'I'm not sure about this and that', they are sending a message to the EU that they can take us for a ride. We have had three years of that. It is time to get this done.
"We gave people a decision. Now parliament is trying to steal it back away from them. When people voted, they voted to leave."
But Ms Rudd dismissed the proroguing of parliament as "the most extraordinary idea I've ever heard" and a "ridiculous suggestion".
The cabinet minister, who is backing Foreign Secretary Jeremy Hunt in the contest said: "My concern is that some candidates are approaching Brexit with a sort of do or die approach and not considering the consequences to people's jobs."
Meanwhile, Tory leadership contender Rory Stewart has said Boris Mr Johnson's plans for Brexit will come "off the rails" once it is subject to scrutiny.
"How is Boris going to deliver Brexit? He keeps saying, 'I am going to deliver it'. I don't even know what he believes. He won't talk to me. He won't talk to you. He won't talk to the public," he told BBC1's The Andrew Marr Show.
Mr Hunt, who came a distant second in the first round, insisted he was the alternative to Mr Johnson.
He told Marr: "The difference between me and Boris, is that I would try for a deal.
"I am not going to create a set of circumstances that makes it all but impossible to get a deal because I think we should be offering the country some better choices."