Labour MPs have been given a fortnight to decide whether they want to stand again for parliament as the party gears up for an early general election by kicking off what some fear could turn into a wave of deselections.

Those MPs who want to continue to represent their constituency in future must first navigate the trigger-ballot process. After a contentious rule-change at last year’s Labour conference, that means commanding the support of two-thirds of local party branches and two-thirds of affiliated trade union branches.

If an MP fails to do so, an open selection process would be triggered in which they could be challenged by other candidates.

Labour’s general secretary, Jennie Formby, wrote to all Labour MPs on Monday, saying they should let the party’s legal department know by 6pm on 8 July “if you wish to remain a candidate at the next general election”.

The letter concluded: “May I also take this opportunity to thank you for the work you have done so far on behalf of the party in parliament.”

A Labour spokesman said: “Ahead of a possible general election, Labour will begin a reselection process of MPs. We are preparing for a general election, whenever it comes,. The process isn’t new.”

Some MPs who have been critical of Jeremy Corbyn fear it could open the way to their removal, though the row over the party’s Brexit stance cuts across many of the usual dividing-lines in the party.

News that the process was being started came before what had been intended to be a significant shadow cabinet meeting, but which senior Labour sources now say is unlikely to shift the party’s policy any closer to a full-throated remain stance.

In parliament on Monday, Corbyn made clear that Labour would try to force the next prime minister to put their Brexit plans “back to the people”.

In response to a statement from Theresa May about last week’s European council meeting,he asked the prime minister: “In her view, what would be worse: crashing out with no deal in October, or putting this issue back to the people for a final say?

“Whoever the next prime minister is, they will barely hold the support of this house, so they certainly have no mandate to force a disastrous hard-right Brexit on this country. Whatever Brexit plan the new Tory leader comes up with, after three long years of failure they should have the confidence to go back to the people on a deal agreed by parliament.”

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Corbyn told colleagues last week he had been reading the autobiography of the party’s former leader Harold Wilson, who advocated remaining a member of the common market in the 1975 referendum while largely taking a backseat during the campaign and allowing cabinet ministers to support either side.

Several shadow cabinet members emerged from last week’s meeting with the clear impression that a decision would be made about Labour’s Brexit stance on Tuesday, after Corbyn had consulted trades union general secretaries.

Corbyn’s deputy, Tom Watson, said: “We had a long discussion last week and we all decided we were going to take a decision at the shadow cabinet. It will be quite a historic day and I hope we can formalise our support for remaining in the EU and supporting the people having the final say on the deal.”

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However, several members of the shadow cabinet remain concerned about any policy that could be portrayed as seeking to overturn the result of the 2016 referendum and Corbyn is now not expected to make a speech on the issue in the coming days.

Tensions at the top of the party exploded into the open after Labour’s disastrous showing in last month’s European elections, with Watson and the shadow foreign secretary, Emily Thornberry, pushing openly for a shift in the party’s stance.

Even some of Corbyn’s most loyal lieutenants, including the shadow chancellor, John McDonnell, have become increasingly convinced that pledging to back a referendum – and campaign for remain – was essential for party morale.

Corbyn said Labour would work across the House of Commons to avert a no-deal Brexit.