MPs are expected to have the opportunity on 29 January to vote on whether to hold a second referendum, after a Conservative backbencher pledged to put down an amendment to Theresa May’s Brexit plan.

Sarah Wollaston had previously withheld her amendment because Jeremy Corbyn would not support it, but she said it was time to put it to a vote of MPs when Brexit was next debated in the Commons, and called on the Labour leader to back her.

Andrea Leadsom, the leader of the House of Commons, announced on Thursday that MPs would again debate Brexit for a full day on 29 January – eight days after the prime minister is due to spell out her next steps in a statement.

MPs will be free to lay a string of amendments to May’s statement, in a debate that is likely to see a range of options put before the Commons for the first time, including Labour’s alternative plan and, almost certainly, an amendment opposing a no-deal Brexit.

Any no-deal amendment would be expected to carry a majority in the Commons, after an amendment submitted by Yvette Cooper to the finance bill earlier this month was carried despite government opposition.

Wollaston’s second referendum amendment, which could attract the support of a dozen rebel Tories, will not be able to pass the Commons unless the Labour leadership backs it.

She said: “I think it’s really important that now we start to test this, and if we’re not successful the first time we could bring it back at a later stage.” The Tory MP indicated she was not certain of success at the first time of asking.

Wollaston added: “I very much hope Labour will finish going through their process and Jeremy Corbyn will come out and stick to their commitment that he will back a second vote, which is what a majority of his members want.”

The Labour leader is showing no immediate signs of backing a second referendum, and on Thursday, Corbyn said his party would submit its own Brexit amendment, calling for a customs union, a close relationship with the single market and enhanced protection for workers and consumer rights after leaving the EU.

Wollaston was speaking at the launch of Right to Vote, a campaign group made up of Conservatives supporting a second referendum, which was attended by half a dozen MPs including Philip Lee, Heidi Allen and Sam Gyimah.

Lee said he believed the second referendum faction in the Conservative party would gradually grow in strength, once other options have been exhausted. “We know cabinet ministers, junior ministers, backbenchers whose head and heart are over here,” the former minister said.

The MP added that May had little room for manoeuvre in the latest round of cross-party talks, because the Tories could split if she moved too far towards a centrist position.

“The idea that we could enter the customs union and the ERG [European Research Group] won’t explode? I don’t see it myself. The Norway model has the potential to split the party much more than a second referendum,” Lee said, although he accepted “we may have to go through that process” of considering each in turn.

The MP also complained that some backbenchers were refusing to come out in support of a second referendum because of fears of violence. “I know of one backbencher who would be here if he had not received a credible death threat,” he said, adding that a large number of colleagues had received similar threats.