The next person to lead the Conservative Party must be a "leading Brexiteer", according to Iain Duncan Smith.
The former Conservative leader said he would be backing someone who "fundamentally believes in Brexit" in any leadership contest.
"A leading Brexiteer has to lead the party and the country out, at the very, very latest October 31, deal or no deal," he told talkRADIO's Julia Hartley-Brewer.
"If we are the party of Brexit we have to go on and do it, and have somebody who genuinely, fundamentally believes in Brexit, and that has been a big mistake."
However, Mr Smith refused to put forward any names he would be backing once Theresa May leaves her leadership post.
"I have pledged myself to believe that we have to get the Prime Minister out before we can start playing games with who is going to do it," he added.
"The best thing for her and everybody else is to break away and say it's time to find a new leader, someone who campaigned for Brexit, someone who is committed to Brexit."
Mrs May is facing increasing pressure to resign today, after Commons leader Andrea Leadsom quit last night over her Brexit Bill.
The Prime Minister announced earlier this week that the Withdrawal Agreement Bill would include "a requirement to vote on whether to hold a second referendum" and a choice over the UK's future customs arrangements.
But Ms Leadsom said she no longer believed that the government’s approach “will deliver on the referendum result” following the announcement.