Labour has not ruled out tabling a motion of no confidence in the government before Christmas if the prime minister fails to bring her Brexit deal back to the House of Commons for a vote.

Tuesday’s weekly meeting of Jeremy Corbyn’s shadow cabinet saw a lively discussion about the best parliamentary tactics in the wake of the prime minister’s shock decision to delay the meaningful vote, the Guardian understands.

The leadership is under intense pressure from grassroots activists and a vocal group of its own MPs, to seek to bring about a general election by tabling a vote of no confidence as soon as possible.

Close allies of Corbyn, including Ian Lavery and Richard Burgon, are cautious, preferring to wait until May has put her deal to a vote, according to sources at the meeting – while the shadow Brexit secretary, Keir Starmer, and the deputy leader, Tom Watson, were keener to press ahead in the coming days.

One shadow minister said May’s statement to MPs next Monday, when she reports on discussions at this week’s European council meeting, would be a critical moment – particularly if she refuses to put her plans to MPs immediately.

Lavery later insisted in a video message: “We’re fully prepared to bring a no-confidence vote when we know we can win,” but said doing so too soon would “serve only to strengthen Theresa May”.

Meanwhile the Unite general secretary, Len McCluskey, a staunch supporter of the Labour leader, published a statement saying Corbyn must not be “bounced” into triggering an early vote.

The shadow chancellor, John McDonnell, told journalists: “We’ll put one down when we can win it. We’ll make a judgment.

“The prime minister’s going off to see what she can get in terms of renegotiations or whatever. We’ll see what she brings back.” He added that Labour would weigh up, “day by day” whether it could win.

Some of those advocating an early move, including the MPs who have signed a letter urging the leadership to table a no-confidence motion immediately, are backers of a second referendum.

They hope a failed no-confidence motion would open the way for the leadership to support a “people’s vote” – though senior Labour sources were keen to stress that would not follow automatically.

The composite motion passed at Labour conference specifies that if May’s deal is voted down in parliament, and the party fails to secure a general election, it “must support all options remaining on the table, including campaigning for a public vote”.

There are also concerns that Labour’s stated strategy – of seeking to take charge of the negotiations and secure a better deal before Brexit day – will look increasingly implausible as the March 29 deadline looms.

Labour’s prospects of winning a confidence vote look highly doubtful, with the DUP suggesting it would continue to support May’s government despite significant reservations about her Brexit deal.

Conservative MP Anna Soubry, one of the most outspoken sceptics of May’s Brexit stance, added to the sense that Labour could not win a vote at this point, saying: “You can be assured when a vote of confidence comes I will be voting in support of my government.”

Labour clashed with the SNP on Tuesday over timing, with the SNP’s leader in the Commons, Ian Blackford, threatening to try to organise his own no-confidence motion.

Blackford said at lunchtime he had an appointment to meet the Labour leader in the afternoon to press the issue in an attempt to speed up to the point where Corbyn’s frontbench would back a second referendum. “Time is of the essence,” Blackford added.

But an hour later Labour sources said the meeting had been postponed.

Parliamentary sources said the SNP could in theory table a motion – but only the official opposition leader can automatically secure parliamentary time for it. The SNP would have to wait for one of its allotted opposition days, of which there are none left this year.

The move by Blackford came after Westminster leaders of the Liberal Democrats, Plaid Cymru and the Green party joined the SNP in calling on Labour to back a no-confidence motion.

A joint letter signed by Vince Cable, Liz Saville Roberts, Caroline Lucas and Blackford was issued on Tuesday morning before all four attended a press conference organised by the People’s Vote campaign.

The letter said: “The government’s inability to pass its Brexit deal through parliament, as witnessed by Theresa May’s withdrawal of her own motion in parliament yesterday, leaves no option for us as leaders of opposition parties but to present a confidence motion on the floor of the House of Commons.”