Theresa May could gain support for her Brexit deal if she promises to stand down as PM, senior Conservatives have told the BBC.
MPs in the party have said they might reluctantly back the agreement if they know she will not be in charge of the next stage of negotiations with the EU.
It comes as newspaper reports claim cabinet ministers are plotting a coup against her.
No 10 has dismissed reports that Mrs May could be persuaded to stand aside.
The prime minister has come under growing pressure to quit following a week in which she was forced to ask the EU for an extension to Article 50, and criticised for blaming the delay to Brexit on MPs.
It remains unclear whether she will bring her withdrawal agreement back to the Commons for a third vote next week, after she wrote to MPs saying she would only do so if there was "sufficient support" for it.
According to reports in some of the Sunday papers, cabinet ministers are planning to oust Mrs May as prime minister and replace her with a "caretaker leader" until a proper leadership contest is held later in the year.