A British Sunday newspaper said Prime Minister Theresa May had signalled in an interview that she might demote Foreign Secretary Boris Johnson, after she faced an open rebellion from within her own party this week.
Friends of Mr Johnson warned Mrs May that she is not strong enough to get rid of him, while Brexiteers said Philip Hammond should be sacked instead.
Another told the newspaper: "Brexit is absolutely crucial to democracy in this country now".
Nadine Dorries, a staunch supporter of Mr Johnson, became the first backbencher to publicly call on the Prime Minister to revive her plan. She is tipped to give an update at 3:30pm local time saying, "I believe we can prove the doomsayers wrong".
The prime minister has successfully survived an attempt from within the Tories to remove her after former Conservative chairman Grant Shapps failed in his attempt to trigger a leadership election.
In the message to Conservative MPs, the Foreign Secretary said he was "fed up to the back teeth" with the reports. I do not know if they are really my friends and allies or if they represent some sinister band of imposters.
Speaking to Sky News, former deputy prime minister Lord Heseltine criticised Mr Johnson.
'I heartily disagree with the sense, tone and spirit of what they are quoted as saying.
Asked about Mr Johnson, Sir Michael added: 'And the foreign secretary'.
She said: "I think Philip Hammond has been deliberately trying to make the Brexit negotiations hard, stall them, obfuscate the issues".
James Airey, Leader of the Conservatives in Cumbria and the Parliamentary Candidate for Westmorland and Lonsdale at June's General Election, said: "In her speech yesterday, Theresa May reached out to a wide audience, just as she did in her inspiring speech on the steps of No.10 past year, and we applaud her".
Former justice minister and fellow Brexiteer Michael Gove also urged Leavers to rally behind May, tweeting: "Strong statement from PM on Brexit - let's be pragmatic over implementation to secure maximum freedom to diverge from European Union in end state".
Mrs Bradley, who served the PM during their time together in the Home Office, dismissed reshuffle talk as "speculation", but delivered a public rebuke to Mr Johnson over his controversial remarks at a Conservative conference fringe event.
'I never discuss whether reshuffles are or are not taking place, ' he said.
Today, May said that the "ball was in EU's court" as the next round of Brexit negotiations got underway.
Speaking to the BBC, Brexit minister Dominic Raab said preparations for a no deal were "well under way", adding that while the United Kingdom had to "strive for the very best outcome" from Brexit negotiations, it must also "prepare for all eventualities".