Labour's tabling of an amendment will see a free vote for the party's MPs.
The amendment calls for a consultation on civil partnerships for same-sex couples to happen immediately, rather than in five years' time as the Government has pledged.
Labour does not want it to jeopardise the passage of the gay marriage bill.
So if the amendment is accepted, the Labour front bench will not back Tory MP Tim Loughton's amendment, which has been condemned by some as a "wrecking" measure.
Yvette Cooper and Ed Miliband will not discuss yet, though, what will happen if the party's proposed alteration to the gay marriage bill is not accepted.