ALBANY  Gov. David A. Paterson said Thursday that he would include gay marriage on the agenda of an extraordinary session he is calling for Tuesday, potentially setting up the first vote on the issue in the State Senate and a dramatic floor debate.

Senate Democratic leaders could still block the issue from coming to the floor and have done so before, though it appears more likely to be put to a vote this time. Gay rights advocates, who feared holding a vote amid a Senate leadership feud this summer, cheered the governor’s action.

The measure’s chances of passing, though, are seen as far less likely. While a same-sex marriage bill has already been passed twice in the Assembly, it faces an uphill battle in the Senate. Democrats have a shaky 32-to-30 edge in the chamber, but several Democrats have said they will vote against the measure, and no Republicans are publicly supporting it.

The primary purpose of the session is to vote on the governor’s plan to reduce a deficit of more than $3 billion, but few expect the governor and lawmakers to agree on a compromise plan by Tuesday. So issues like same-sex marriage and measures to reform the pension system and public authorities are likely to dominate the session.