The Labour Party plans to give MPs the opportunity to vote on the European Arrest Warrant (EAW) on the day before the hotly contested Rochester and Strood by-election.
The move comes after the government re-signed the UK up to 35 European Union justice measures, including the controversial warrant.
The government was criticised by MPs, including Conservative backbenchers, because parliament was denied a vote on whether to re-join the controversial arrest warrant.
Instead, the Prime Minister David Cameron said MPs could vote on just 11 relatively minor measures.
The Speaker of the House of Commons, John Bercow, said the decision could be seen as "contemptuous" by the public.
"Most of us think that a commitment made is a commitment that should be honoured, and we should try to operate according to sensible standards, rather than try to slip things through some sort of artifice," Bercow said.
"It may be the sort of thing that some people think is very clever, but people outside of the house expect straightforward dealing.
"And they are frankly contemptuous, and I use the word advisably, contemptuous of what is not straight dealing. Let's try to learn from this experience and do better."
The Shadow Home Secretary, Yvette Cooper, then called for a vote on whether the issue should be debated.
The Labour Party narrowly lost (272 to 229) the vote in the House of Commons after Conservative Party whips frantically got their MPs into the chamber.
Even the Prime Minister reportedly raced back from a speech at Mansion House to place his name on the record.
Cooper has now announced that she wants to use the next opposition day debate, on 19 November, to allow MPs to vote on the EAW.
"The House of Commons should be able to vote to endorse the European Arrest Warrant, putting it beyond legal challenge – something we know Eurosceptics have been planning," Cooper said.
"Despite being promised this vote, MPs have been denied the chance to give overwhelming backing to this crucial crime fighting measure.
"The government must put that right at the earliest opportunity.
"May was very foolish not to listen to the strong support across the House both for the EAW and for the opportunity of a comprehensive vote.
"She has handled this dreadfully. It's a complete mess."
The vote in the House of Commons would come before the Rochester and Strood by-election on 20 November, where Ukip are tipped to win.