Michael Gove has refused to comment on his cabinet future.
The Education Secretary dodged reporters' questions as he left his home this morning.
Reports have indicated that Mr Gove will not resign, however he has refused to confirm his position publicly.
Former Brexit secretary David Davis - who quit in July over Theresa May's Chequers plan for leaving the EU - said it was still possible to reopen negotiations with Brussels.
"It is a dreadful proposal, it really doesn't fly by any measure. It is not a deal we should accept," he told the BBC Radio 4 Today programme.
"The European Union has spun this out deliberately to try to use time against us.
"It will get to a point, I am quite sure, when it will be very tense and people will be very nervous about the way the negotiations go, but European negotiations are never over until they are concluded."
A former permanent secretary at the Foreign Office, Sir Simon Fraser, has dismissed Mr Davis's claim that it was still possible to reopen negotiations with the EU.
"David Davis was a terrible Brexit secretary. He could hardly be bothered to go to Brussels and rapidly lost respect there," he wrote on Twitter.
"Preposterous for him now to suggest that EU deliberately delayed negotiations. They spent months waiting for him to engage."
Housing Secretary James Brokenshire has said he "very much" wants Michael Gove to continue in office, amid reports the
Environment Secretary is considering quitting over the Brexit deal.
Asked if Mr Gove was still a member of the Government, Mr Brokenshire told the BBC Radio 4 Today programme: "Everything I've heard in terms of... yes, Michael is a great colleague.
"I have enjoyed working with him and want to continue to work with him in his role as Secretary of State at Defra.
"I very much want to see him continuing in that role and I wish him well for the future in dealing with that."
Mr Brokenshire dismissed a claim by David Davis that the Government could still reopen negotiations on the withdrawal agreement with Brussels.
"The very clear and firm assessment is that this is the deal that needs to be concluded," he told the Today programme.
"It strikes a fair balance. Yes, there are compromises that are there. But there is no sense that there will be anything more that can be gained through further negotiation.
"Indeed, it may well take us backwards if you were to try and even do that."
Mr Gove remained tight-lipped when asked if he would resign as he arrived at the Department for Environment, Food and Rural Affairs in Westminster.
Shadow chancellor John McDonnell has claimed Labour could secure a Commons majority for a "compromise" Brexit deal.
Mr McDonnell said a "unity platform" was emerging at Westminster to avoid the "catastrophic" impact of a no-deal break with the EU.
"I think we can secure a majority," he told the BBC Radio 4 Today programme.
"People have looked over the edge of a no-deal Brexit and realised it could be catastrophic for our economy.
"Our European partners, I think, also have looked over the edge of a no-deal Brexit and seen what an impact it could have on their economies.
"So I think what is emerging within the House of Commons now is almost a unity platform to avoid a no-deal."
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