Pro-remain MPs are pulling back from plans to force a vote on attaching a second referendum to Boris Johnson’s Brexit deal on Saturday owing to fears that they do not have enough support.

Leading supporters of a “people’s vote” had been planning to use so-called “Super Saturday” to test support for a second referendum. But sources close to the campaign said there was not enough backing among former Tories for the move to be successful. They denied that their pulling away from a vote was because the Labour leadership were lukewarm on the idea.

As recently as Wednesday night, Jenny Chapman, a shadow Brexit minister, had said she was sure there would be a second referendum amendment and that Labour would support it, but Jeremy Corbyn has been less enthusiastic.

Pro-remain MPs insisted it was a better idea to hold the vote at a later stage of the process, as soft Brexit Tories will be more likely to support a second referendum in the event that Johnson’s deal fails.

In this scenario, the opposition could seize control of the order paper via an SO24 debate and hold a vote on a second referendum. If the deal does pass, they could seek to amend the withdrawal agreement bill before exit day.

However, the lack of movement on Saturday is likely to disappoint many grassroots campaigners for a second referendum. Hundreds of thousands of protesters are expected to turn out in Parliament Square on Saturday.

Expectations had been rising that Labour would work together with other opposition parties to try to secure a referendum. Chapman told the BBC’s Andrew Neil on Wednesday evening: “Should a deal be tabled on Saturday, I’m as sure as you can be that there will be an amendment tabled that would want to see a referendum attached to the deal. I would expect us to support that.”

However, when Corbyn was asked on Thursday whether Labour would support such a bid, he suggested his first priority was to vote down Johnson’s deal. “We are unhappy with this deal and as it stands we will vote against it, although obviously we will need to see all of the last details of it,” he said.

On the deal more broadly, he told Sky News: “From what we’ve read of this deal, it doesn’t meet our demands or expectations, it creates a border down the Irish Sea, and it leads once again to a race to the bottom in rights and protections for British citizens and a danger of the sell-off of our national assets to American companies.”

Asked whether he had given up on his plan to negotiate a Labour Brexit deal, he replied: “Not at all. A Labour government elected in a general election would within three months negotiate an agreement with the EU around the five pillars … and then within six months we would put that to a referendum.”

Corbyn expressed caution last weekend about the idea that Labour could support any Johnson deal in exchange for a referendum.

The original plan advanced by proponents of a “confirmatory vote”, including the Labour MP Peter Kyle, was that MPs would pass an amendment making the Brexit deal conditional on a referendum, and then once it was amended they would vote with the government for the deal.

But a number of Labour MPs have concerns about anything that would appear to be offering support to Johnson, albeit conditionally. And a potentially larger group have concerns about supporting a referendum at all. When Labour recommended that its MPs back the idea of a referendum in the so-called “indicative votes” process earlier this year, 27 rebelled to vote against it. Several more abstained.

If Johnson’s deal fails and he is forced to request an extension, it is possible he could move to call an election as soon as Monday. Labour would find this hard to resist as Corbyn has repeatedly said he is ready to vote for one as soon as the prospect of a no-deal Brexit is off the table.

The SNP tabled an amendment to Johnson’s deal motion on Thursday calling for a three-month extension to article 50 in order to hold a general election, indicating that the party would also support one.

The Liberal Democrats have been in favour of a second referendum before any election. However, the party will not be tabling an amendment attaching one to Johnson’s deal, even though their leader, Jo Swinson, said: “The fight to stop Brexit is far from over. When this deal comes to parliament we will use every possible opportunity to give the public a people’s vote on the Brexit deal that includes the option to remain in the European Union.”