Jeremy Corbyn faces up to 10 resignations from the Labour frontbench if he fails to throw his party’s weight behind a fresh attempt to force Theresa May to submit her Brexit deal to a referendum in a fortnight’s time, frustrated MPs are warning.

With tension mounting among anti-Brexit Labour MPs and grassroots members, several junior shadow ministers have told the Guardian they are prepared to resign their posts if Corbyn doesn’t whip his MPs to vote for a pro-referendum amendment at the end of the month.

Corbyn has been struggling to balance the conflicting forces in his party over Brexit, as the clock ticks down towards exit day on 29 March..

Many party members and MPs would like him to take a lead in seeking to block Brexit before time runs out – but some frontbenchers are equally adamant they could never support a referendum.

Len McCluskey, the general secretary of the Unite union and a close ally of Corbyn, risked stoking the conflict in the party on Wednesday when he argued that stopping Brexit was “not the best option for our nation”.

“My view is that, having had a 2016 referendum where the people have voted to come out of the EU, to try and deflect away from that threatens the whole democratic fabric on which we operate,” he told Peston on ITV. “I’m saying that in reality it is not the best option for our nation.”

The party’s conference policy, thrashed out in a late-night meeting chaired by the shadow Brexit secretary, Keir Starmer, suggests that if it has failed to secure a general election “Labour must support all options remaining on the table, including campaigning for a public vote”.

And it includes the line: “If the government is confident in negotiating a deal that working people, our economy and communities will benefit from they should not be afraid to put that deal to the public.”

Since the prime minister’s deal was overwhelmingly rejected by parliament last month, Labour has continued to engage with the government after Corbyn initially declined a meeting with May.

The pair exchanged letters last week setting out their respective Brexit positions, and senior Labour figures met ministers on Tuesday.

Starmer and the shadow Cabinet Office minister, Jon Trickett, were accompanied by Corbyn’s close advisers Seumas Milne and Andrew Fisher as they met the Brexit secretary, Steve Barclay, and May’s de facto deputy, David Lidington.

A Labour party spokesperson said: “Keir Starmer and Jon Trickett had a frank and serious exchange with Stephen Barclay and David Lidington following Jeremy Corbyn’s five-demands letter to Theresa May.

“Starmer and Trickett set out Labour’s five demands and pushed the government to change its red lines.” Senior Labour sources said they believed the meeting had gone well, and expect a follow-up session to be held next week. But anti-Brexit MPs believe it is time to shift the policy towards supporting a referendum, as May has rejected Labour’s advances by continuing to rule out a customs union.

Some see Starmer as the most likely champion for their cause inside the shadow cabinet. Other key figures, including Trickett and the Labour chair, Ian Lavery, are more sceptical.

Attention is focusing on plans made by the Labour MPs Phil Wilson and Peter Kyle, under which parliament’s endorsement of a deal would be made subject to the public’s approval, echoing the model pursued for the Good Friday agreement.

“That’s where the action is,” said one backbencher. There could also be a straightforward “people’s vote” amendment when May brings an amendable vote to MPs on 27 February.

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Several frontbenchers, including the shadow Treasury minister, Clive Lewis, and the shadow business minister, Chi Onwurah, will speak at a Love Socialism, Hate Brexit rally at parliament before the vote on Thursday. The rally is being promoted by the pro-referendum group Another Europe is Possible.

Lewis previously quit as shadow defence secretary in order to rebel against the triggering of article 50, but has since returned to the Labour frontbench.

He said last week: “I’m on the frontbench because I live in hope that the party will get to the bit of our conference policy where it supports a people’s vote.”

Even some shadow frontbenchers opposed to a second referendum are coming to the view that the party needs a “cathartic moment” to whip for a new referendum and prove that the plan has no parliamentary majority, with more than 20 Labour MPs likely to oppose. One sceptical shadow minister said the February deadline would be the time for “peak agitation”.

Tensions over the party’s Brexit policy have been simmering since Corbyn made his offer of a Brexit compromise in a letter to May last week, prompting former leadership candidate Owen Smith to suggest he could quit the party if Labour eventually backed a Brexit deal.

Anger spilled into the open on Wednesday. The MP Neil Coyle claimed the party was losing members and councillors, and could yet lose MPs over its Brexit policy.

Neil Coyle (@coyleneil) I can't make Parliamentary Committee today & as emails are screened I thought I'd tweet the boss instead. So @jeremycorbyn here goes. Members leaving in their thousands over Brexit. Cllrs quitting. MPs will leave. Antisemitism continues in your name. Only you can change all this

One Labour MP, Geraint Davies, tabled an amendment attempting to bring forward the crunch point for the Labour leadership by calling for a referendum on May’s deal but it was not directly connected with that of Kyle and Wilson. “It’s very premature,” one MP said.

Davies’ amendment, which he submitted on Wednesday, will be attached to a government motion laying out the next stage of Brexit negotiations. MPs are due to vote on the motion on Thursday.

Davies said his amendment would help build support for another legally binding amendment on a second referendum at the next opportunity.

“There is a hidden majority in the house that want to support this position, but they are all waiting around for some guiding north star for them to follow before they take action, but we are running out of time,” he said. “MPs are in danger of being complicit if they do not take action now.”

But Kyle said he was not behind Davies’ amendment and had hoped to wait until there was more momentum behind the move.

“The main push for this will be the meaningful vote. Clearly the landscape changes but this vote tomorrow is not binding and this amendment should not be interpreted as the main bid for support,” he said.

Corbyn was criticised for failing to take tougher action against MPs who declined to back Yvette Cooper’s amendment aimed at averting a no-deal Brexit; but chief whip Nick Brown has made clear to MPs he would deal with all Brexit votes in the round, at the end of the process.