Jacob Rees-Mogg has said that he would be willing to back Prime Minister Theresa May's Brexit withdrawal deal if the DUP does likewise.
Writing in the Daily Mail on Wednesday Mr Rees-Mogg apologised for changing his mind on the deal, saying those who negotiated it are guilty of a "failure of statesmanship".
The European Research Group chair said that he believed Mrs May's deal was now the only way to achieve Brexit.
However, the DUP still seem no closer to backing the deal, with their Brexit spokesperson Sammy Wilson saying on Tuesday that he would prefer a one year extension to the "toxic deal".
"There are some colleagues who I admire greatly and who have stood firmly with us in defending Northern Ireland who now take the view that the Withdrawal Agreement, even though it is a rotten deal, is better than losing Brexit," Mr Wilson said.
"To them I say that, if the deal goes through, we have lost our right to leave the EU."
Speaking on Wednesday morning Mr Rees-Mogg said that, while the deal is "deeply unsatisfying", it is the only way left of leaving the EU.
"We are in a terrible constitutional muddle, we have a broken-down relationship between the executive and legislative," he said.
"In those circumstances, the deal that Mrs May has got, with its many faults, is at least leaving the EU.
"I haven't changed my mind on the deal. The only way left of doing it (leaving the EU) is this deeply unsatisfying deal."
Asked if the DUP is going to support Mrs May's deal, the North East Somerset MP said "you'd better ask the DUP".
"The union of the United Kingdom is the most important union," he said.
Mr Rees-Mogg said he could not speak for other members of the ERG.
He said there are a number of things that have to fall into place if the deal is going to pass.
"Mostly the DUP - they are still against it," the ERG chair said.
"If they shift, I think the chances of getting the deal through are pretty good."
The support of the DUP and the hardline Brexiteers in the ERG are seen as key for Mrs May if she wishes to get her Brexit deal through the Commons.
While some in the ERG may be beginning to waver, it appears as though the DUP position remains unchanged and they will not back Mrs May's deal during a third Meaningful Vote if it takes place later this week.
MPs are set to hold a series of indicative votes in the House of Commons on Wednesday to test support for other Brexit proposals, after Parliament seized control of the agenda from Mrs May on Monday.
Mr Rees-Mogg said the alternative Brexit votes scheduled for Wednesday are about MPs trying to thwart Brexit.
"We ought to be leaving on Friday at 11 o'clock with or without a deal but that's not the situation we find ourselves in," he said.
"They (MPs) didn't like the result, they thought the British people have got it wrong, that's what today is all about."