Theresa May will fly to Brussels on Saturday to seal her Brexit deal with EU leaders, but the real hard work will begin on her return, when she must win back dozens of her own MPs to get the agreement through parliament.

Eighty-five of the Conservative party’s 315 MPs have publicly rejected the controversial Brexit settlement, an analysis has shown. The prime minister would have to win over nearly an equivalent number of Labour rebels to get the final deal approved, a task considered sufficiently unlikely that May faced questions on Friday about whether she would resign if MPs voted against it.

She repeatedly refused to rule out the option, telling a phone-in on BBC Radio 5 Live: “I’m not thinking about me. I’m thinking about getting a deal through that is good for the country. My focus is on getting this deal through.”

The phone-in was the latest step in what has become a punishing media blitz as May and other senior ministers try to sell the deal directly to the public – many of whom are weary of the Brexit debate – in the hope that a change in the national mood will pressurise MPs into supporting her.

May is expected to announce details of the government’s post-Brexit migration plans in the run-up to the crucial vote, including lifting the cap on highly skilled “tier two” migrants such as doctors.

Leaked cabinet papers obtained by the Telegraph suggest the government is also considering opening up a new route for low-skilled migration to ease the impact on the economy of the end of free movement. Proposals under discussion include 11-month work visas and a special category for 18-30-year-olds who want to work in the UK for up to two years.

Senior government sources suggested that Downing Street had initially been sceptical about the need for lower-skilled migration routes, but was wary of alarming business groups.

Most routes for low-skilled migrants will stop, and May is likely to sell this much more stringent regime as a vindication of her claim that her Brexit deal will end free movement. She has suggested that she believes the desire to control migration was a key rationale for the referendum result in 2016.

The prime minister was criticised earlier this week when she told the CBI that EU migrants would no longer be able to “jump the queue”.

Jeremy Hunt, the foreign secretary, said in a live chat on Facebook that there would be a constitutional deadlock if parliament rejected the deal. “I think there would be a huge amount of worry by many people who just want the government to get on and deliver Brexit,” he said.

May will meet the EU commission president, Jean-Claude Juncker, on Saturday before a summit with EU leaders on Sunday, at which the withdrawal agreement and political declaration are expected to be formally adopted after almost two years of negotiations.

The deal will then have to be debated and voted on by British MPs in a “meaningful vote”, a step wrung from the government last year during the passage of Brexit legislation.

The government is likely to announce the date of the vote on Monday, though it is expected to be in the first two weeks of December.

Downing Street has launched a “Brexit Facts” blog aimed at rebutting negative press coverage of the deal and May’s spokeswoman said on Friday: “You will see ministers engaging widely”.

Philip Hammond, the chancellor, will visit Belfast to appear at the annual conference of the Democratic Unionist party, whose 10 MPs refused to support the government on finance bills this week because they do not like May’s Brexit deal.

The DUP leader, Arlene Foster, warned on Friday that the party’s confidence and supply agreement with the Conservatives which is propping up May’s administration would be at risk if the Brexit deal was passed.

She told BBC Radio Ulster’s Inside Politics: “If she is successful in parliament, and there is no evidence that she is going to be successful, then of course we will have to revisit the confidence and supply agreement. That agreement was about giving national stability; it was acting in the national interest and delivering on Brexit. If this is not going to deliver on Brexit then, of course, that brings us back to the situation of looking at the confidence and supply agreement, but we are not there yet and we are not going to jump ahead until we see what happens in parliament.”

The leading Brexiter Boris Johnson, who has repeatedly criticised the deal, will also be appearing at the DUP conference. Allies of Johnson said the former foreign secretary intended to increase his criticism of May’s plan in the coming weeks, having repeatedly warned that it would lead to an unacceptable loss of UK sovereignty to the EU.

In a sign that May is prepared to deploy her powers of patronage to help get the deal through, it was announced that the Conservative backbencher John Hayes would receive a knighthood. Hayes is a Brexiter, but has not submitted a letter of no confidence in the prime minister or publicly declared whether he would support her deal.

Despite the level of opposition in parliament, May suggested MPs could change their mind if they listened to the electorate. “In a sense, I think there’s two conversations going on at the moment,” she said. “In parliament there’s a lot of focus on who’s going to vote for the deal or not, and outside I think people are thinking: ‘Actually, let’s make sure we can get this through and get on with delivering’.”

May played down the idea of no Brexit, which some ministers have begun to suggest is the most likely alternative to the deal, as the People’s Vote campaign wins the backing of a growing number of MPs. “Personally, there’s no question of no Brexit, because the government needs to deliver on what people voted for in the referendum in 2016,” she said.

However, Dominic Raab, who resigned as Brexit secretary last week in protest at the withdrawal agreement, suggested that the government deal would be “even worse” than staying in the EU.

Asked on BBC Radio 4’s Today programme what he would do if he had to choose between May’s deal or no deal, Raab said: “Well, I don’t have to choose that. I’m sorry, I’m not going to give way to hypothetical scenarios. I’ll keep fighting for the best, most successful Brexit.”

Pressed further on whether he thought the deal would be worse than staying in the EU, he replied: “Well, I’m not going to advocate staying in the EU but if you just presented me terms, this deal or EU membership, we’d effectively be bound by the same rules without a control or voice over them, [so] yes, I think this would be even worse than that.”

In recent days, Brexit-backing ministers who have stayed in May’s cabinet have been less publicly supportive of the deal than remainers, such as Damian Hinds and Matt Hancock.

Friends of Andrea Leadsom said she had no plans to campaign for the agreement and allies of Michael Gove suggested he would be willing to promote those aspects of the deal that cover his brief of fishing and farming, but not the wider package.

Gove turned down the job of Brexit secretary last week after May rejected his suggestion that the withdrawal agreement be renegotiated to alter the terms of the Irish backstop.

Meanwhile, Leadsom is keen for the government to consider a “no deal plus” if parliament votes against the deal. Under that plan, the government would pay the EU £20bn – less than half the planned divorce payment – in exchange for a two-year transition period, during which the UK would prepare for operating on World Trade Organisation terms.