Senior Conservatives have offered Theresa May a glimmer of hope by rallying behind attempts to modify her Brexit plan, amid hints the Democratic Unionist Party (DUP) could soften their opposition.

Sir Graham Brady, the influential Tory backbench organiser, has backed a bid to salvage the prime minister’s deal by removing the controversial backstop element, a move also supported by Ms May’s long-time ally Damian Green and a group of Tory grandees.

The new amendment, which would see the backstop replaced with “alternative arrangements”, would compel Ms May to return to Brussels to seek changes to her deal, if backed by MPs in the Commons on Tuesday.

It comes amid reports the DUP has privately decided to support Ms May if there was a time limit on the backstop plan, with party sources suggesting to The Sun that their staunch opposition to her Brexit deal could be weakening.

Sir Graham, who voted against Ms May’s deal, told The Independent that removing the “unacceptable” backstop would leave a plan that the “majority” of Tory MPs could support.

He also said the DUP could “live with” the deal if the backstop was gone.

Sir Graham, the 1922 committee chairman, said: “My main purpose is to try to demonstrate that the Conservatives and DUP, our confidence and supply partners, can support something and demonstrate there is a position that can command a majority in the House of Commons.

“I think most MPs would like an agreement as we leave the EU, rather than no agreement and WTO [World Trade Organisation] terms.

“This is the only agreement – as long as the backstop is removed – that can command a majority in the House of Commons.”

DUP Westminster leader Nigel Dodds said the “trap of the backstop” was the main sticking point and suggested the EU was shifting towards a compromise on the issue.

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He said: “Whether it is the Lithuanian, Polish or indeed the EU Commission’s chief spokesman, there have been several signals that people are prepared to look at other options. That is positive as it shows people are starting to be more realistic.

“We continue to work with our government in an effort to reach a better deal which can command support in the House of Commons but also importantly in Brussels too.”

Mr Dodds added: “The trap of the backstop is the problem. There are ways forward which do not require this backstop and we need to see a willingness to explore such options.”

The amendment was tabled on Thursday night by senior backbencher Andrew Murrison and it has been backed by a number of influential figures in the Tory party machine, including 1922 executive committee members Cheryl Gillan, Charles Walker, Bob Blackman and Nigel Evans.

MPs are scrambling to come up with amendments to the prime minister’s Brexit blueprint after the Commons overwhelming rejected her deal by a record 230 votes earlier this month.

Labour’s Yvette Cooper has teamed up with Tory MP Nick Boles to try to rule out a no-deal Brexit, while former Conservative attorney general Dominic Grieve wants a series of “indicative votes” to test parliamentary support for different scenarios.

However Ms May will still have to convince Brussels to back whatever MPs support, which represents a major challenge.

French finance minister Bruno Le Maire warned that the Irish border backstop issue is “done” and would not be revisited.

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He said the EU has “nothing to give” the UK on the Brexit deal apart from “clarifications” and it was up to the UK government to find a way out of the situation it found itself in, not EU member states.

Mr Le Maire, who is at Davos, told the Today programme: “Reopening the deal would mean weeks, months, of new negotiations between the UK and the European member states – that is exactly what we want to avoid.”

Asked about whether no-deal measures taken by the EU would be as bad as some groups have suggested, including border delays between Dover and Calais, Mr Le Maire said: “We are aware that a no-deal situation would have concrete consequences and would force us to take some decisions to reinforce the control at our borders.