Theresa May will be within 20 votes of winning a parliamentary majority for her Brexit deal if she can gain assurances from the EU that will persuade the Democratic Unionist party to back her deal, senior ministers and Tory MPs believe.

One cabinet minister said they believed the success of the prime minister’s deal hinged entirely on a last bid to win round the DUP. Another MP said they saw the Northern Irish party as the “British standard” who would give them the reassurance they needed to fall in behind.

“You unlock huge numbers of Tory MPs if you can get something the DUP can accept,” the cabinet minister said. “There’s no point at all in holding a vote until you win back the DUP. That is the absolute priority.”

Several cabinet sources played down the prospect of any efforts to try to form a coalition of support with Labour MPs and said all efforts were focused on regaining the DUP’s support.

“You cannot get this deal through only on the back of Labour votes because it would split the Tory party,” one official said. “That means one thing – bringing the DUP back on board.”

Sources said they had also been encouraged in recent days by the more conciliatory noises being made by Eurosceptic Tories after Labour attempted to hold a no-confidence vote in the prime minister.

May made great play of a renewed relationship with the DUP when she addressed the 1922 Committee of backbench Tory MPs last week, telling them she had made mistakes and allowed the relationship to “drift” but that she and Arlene Foster, the DUP leader, were now on “one page”.

The view is similar in Brussels and Dublin. One senior EU source said May’s chances of getting talks reopened in Brussels were doomed until she could communicate to EU negotiators exactly what the DUP will accept in terms of assurances over the Irish backstop.

“There is no point in giving any assurances or reopening negotiations if there is no chance of getting a deal approved in the house,” the source said. “She has to go back and find out what the DUP will accept. If that is no backstop then we are in no deal, if it is not, then we can talk.”

One former cabinet minister, who has been highly critical of May, said they believed the prime minister may be within “20 or 30 votes” of getting her deal through parliament, depending on whether she could regain the support of the DUP. This would bring on board all but the most hardline Conservative Brexiters.

They also said the Irish taoiseach, Leo Varadkar, could break the impasse with some minor concessions, suggesting he had to budge about 5% to get the deal across the line. “That’s all he needs to do.”

While the EU has been adamant negotiations on the withdrawal agreement cannot be reopened and the backstop must remain, May has indicated that officials are looking at a new solution that would remove the need for the backstop.

She told the House of Commons on Monday she would be seeking legal and political assurances around “the firm date for introducing the future relationship”.

“That is currently December 2020, we will be continuing discussions on this point,” she said.

One MP said this was a “creative idea” because if the new start date was moved to not later than December 2022, this would dovetail with the two-year transition period extension option in the current deal.

But one DUP source said: “The difficulty is how can you legally guarantee trade talks will be over by any date?”

On Tuesday, the DUP MP Gavin Robinson said the party was engaging constructively with the changes May was pursuing. “She went back to the European commission, she has said that work is ongoing and that the changes will be legally enforceable,” he said.

“We are working with the prime minister, she has indicated she will get these assurances and we will assess them at the time. I think we are constructively engaging in the aspiration that we have, and that the majority of parliament has, which is that we need a legally binding assured amendment to the withdrawal agreement.”

However, Robinson said there was “little comfort in the documents shared over the weekend” after May attended the Brussels summit of EU leaders.

Privately, DUP sources were more sceptical of the progress being made, saying “tinkering around the edges” would not be enough and that the party remained unconvinced May could extract a guarantee over a firm start date for a future trade deal.

It is understood there are no meetings scheduled over Christmas between May and DUP leaders.

There are also concerns in the DUP that anything agreed now that does not have legal standing could be brushed aside by future leaders. “We don’t know what personalities or parties are going to be in power in five years,” one party source said.

The European commission’s chief spokesman this week ruled out talks in the near term. “The deal that is on the table is the best and only deal possible, we will not reopen it, it will not be renegotiated,” he said. “The European council has given the clarifications that were possible at this stage so no further meetings with the UK are foreseen.”

Cabinet sources suggested they were quietly hopeful that MPs might come back from the Christmas recess with a more positive view of the deal when parliament votes on 14 January. Although one said they doubted MPs would fold on their decision to vote down the deal until they had defeated it at least once.