MPs have “a duty” to pass Theresa May’s Brexit deal in the House of Commons and ensure the UK leaves the EU, the health secretary has said, as the prime minister and her team prepared for a final push to persuade MPs to back it.

In a round of broadcast interviews on Monday morning, Matt Hancock insisted the long-awaited withdrawal agreement bill (Wab) was both a new measure and the only way to deliver on the referendum result.

“It ultimately will come down to this when MPs are voting: do you want to deliver on the referendum result? Not, is this your perfect resolution to Brexit, and exactly what you want, but this is the piece of legislation that would deliver on the referendum,” he told BBC One’s Breakfast programme.

“And I think, therefore, as I believe in democracy, we have a duty to deliver it.”

Speaking later on ITV’s Good Morning Britain, Hancock said: “Ultimately for me, this is about delivering on promises, and parliament has to step up.”

May will ask her cabinet on Tuesday to sign off a package of Brexit concessions in a final attempt to push a package through the Commons, most likely in the week starting 3 June.

With the Conservatives on course for a drubbing in the European elections on Thursday, the prime minister hopes the results will focus the minds of her MPs and persuade them to support the bill.

Despite the collapse of cross-party talks with Labour, ministers hope some of the measures discussed can still be bolted on to the bill, as part of what May has called a “new, bold offer to MPs across the House of Commons”.

While the widespread prediction is that the bill will be heavily defeated, Hancock said critics should hold their fire. “They haven’t seen the proposals. The proposals will be discussed in cabinet tomorrow, and then published,” he said.

He insisted the Wab was not the same as the departure plan heavily voted down by MPs before. “That is different from the actual legislation that brings forward the agreement to leave the European Union, which includes in it a whole load of proposals for what the future relationship is, as well as details of the actual withdrawal agreement,” Hancock said.

Proposals are expected to include separate legislation to ensure parliament is given a vote on whether to adopt any improvements to workers’ rights introduced by the EU27 in future – though that would fall short of Jeremy Corbyn’s call for changes to be automatically adopted.

The government is also keen to offer fresh reassurances to the Democratic Unionist party, which has been resolutely against May’s deal and is particularly concerned about the risk of regulatory divergence between Northern Ireland and the rest of the UK.

The plan would, Hancock said, show the government had “engaged with a huge amount of people, right across the house”.

In a veiled warning to would-be successors to May who would want to change the approach to Brexit, Hancock noted even a new leader would face the same parliamentary arithmetic.

“The only other way to get a different sort of Brexit … would be to have a general election. And to have a general election before we have delivered on Brexit would ultimately go to the heart of the failure so far to deliver on commitments,” he said.

Hancock, who is expected to be among a crowded field seeking to take over from May, told BBC One he was “not going to rule it out”.

Several cabinet ministers, including the Brexit secretary, Steve Barclay, are likely to press for the government to ramp up no-deal Brexit preparations, in case May’s deal is defeated yet again.

“Members of parliament do need to face facts, and if the deal were not to go through then there are only two alternatives … You either leave with no deal or you revoke,” he said.

“If parliament won’t back a deal … I do think we need … to bring forward our preparations to mitigate no deal, because we will need to use the additional time we have, and we need to move at pace to do so.”

Cabinet ministers keen on a softer Brexit prefer the idea of holding a series of votes in parliament before the bill is tabled, a process that could reveal a majority for a customs union – though that was not the outcome the previous time indicative votes were held.

As well as the substance of the government’s “bold” offer, May must decide on when the crunch vote on the bill will be held.

Downing Street has committed to the week beginning 3 June, but with Donald Trump and a string of world leaders visiting the UK that week to mark the anniversary of D-day, timing is tight.

If the government hopes to hold the vote at the start of the week, before the US president arrives, it would face pressure to publish the bill this week, before MPs disappear for a Whitsun recess.

But that could amplify objections to the government’s policy as voters prepare to head to the polling stations for European parliament elections.