Theresa May is facing her first Commons defeat on her Brexit bill within days, amid a last-minute plea to pro-European Tories to force a “truly meaningful vote” on Britain’s final EU deal.

Leading Tory rebels believe they have a real chance of securing the measure, with party whips also said to be worried that they cannot prevent an embarrassing defeat for the prime minister before she heads off to Brussels for a historic EU summit.

In a joint statement on Saturday night, senior figures from across the Commons told the Observer that MPs considering whether or not to join this week’s rebellion should “put the national interest, not party politics, first”.

The statement is signed by the leaders of the all-party parliamentary group on EU relations, including former Tory business minister Anna Soubry, Labour’s Chuka Umunna, Jonathan Edwards from Plaid Cymru, Stephen Gethins from the SNP, Caroline Lucas from the Greens and Jo Swinson from the Lib Dems. They warn that ministers may be quick to use “exceptional powers to take the UK out of the EU on a bad deal, or with no agreement at all”.

“It has been said that in the EU referendum last year, the British people voted to ‘take back control’ of our laws,” they write. “For many, that meant a powerful reassertion of parliamentary sovereignty. As such, we parliamentarians have a solemn duty to scrutinise the actions of the executive. On no subject is this scrutiny more important and more necessary than Brexit.

“Members of all parties have already provided valuable scrutiny to the EU (Withdrawal) bill, and we have forced the government into some concessions. But little of that will matter unless we can have a truly meaningful vote on the withdrawal agreement the government negotiates with the European Union.”

The warning comes as a poll suggests that most voters back handing MPs a final say on the deal. A BMG poll for the political website Left Foot Forward found that 61% backed giving MPs such a vote, including 52% of the over-55s and 39% of Tory voters. Most (58%) also believed MPs should have a final say on the EU divorce bill. The poll was conducted online among 1,500 people between 5-8 December and was weighted to reflect the population.

The “meaningful vote” amendment was drawn up by the former attorney general, Dominic Grieve, and calls for MPs to have a binding vote on the final Brexit deal before ministers can formally enact Britain’s departure from the EU. Those backing the move believe it would give MPs a greater chance to amend the Brexit deal, forcing ministers back to the negotiating table.

Many Tories had expected ministers to present a compromise deal over so-called amendment seven, but none has been forthcoming. A major verbal concession could head off the risk of defeat. The group of MPs call on their colleagues to “stand up for the sovereignty of parliament and the people we were sent here to represent”. “Ministers have committed to enshrining the withdrawal agreement in legislation which must be passed through parliament,” they write. But there are no guarantees that this would be in any way meaningful. Were the government to be defeated, they could still crash the UK out of the Union with no deal. There is no prescription for how much detail must be provided in the bill. And we could even find ourselves in the absurd position of voting on the exit agreement after the UK has left.

“That is why we believe it to be vital that colleagues from across the House vote on Wednesday to give parliament a properly meaningful say on Brexit by supporting Amendment 7. It helps ensure a meaningful vote comes before exit day and that the terms of our withdrawal should be approved by way of a legally binding Act of parliament, which should be passed before the prime minister signs any agreement.

“We understand the pressures colleagues face to vote ‘the right way.’ But this is a matter of the national interest, and that interest must come first.”