New York (CNN) -- The New York Senate failed to bring to the floor a bill that would legalize same-sex marriage Monday, but talks will continue Tuesday, according to a spokesman for Senate Majority Leader Dean Skelos.
Monday was technically the last official day of the legislative session, but votes are still pending on two other major pieces of legislation, one dealing with rent control in New York City and one on a property tax cap for the whole state.
Large groups of protesters on both sides of the issue lined the halls of New York's Capitol on Monday.
The bill, if approved, could pave the way for New York to become the nation's sixth state to legalize same-sex marriage.
A vote on the measure, which the state Assembly passed Wednesday night, has been stalled in part by Republican concerns over protections for religious institutions against the potential for litigation in the wake of the proposed law.
Republicans, led by Skelos, expressed concerns Friday over the "unintended consequences" of a bill that redefines the legal parameters of marriage in what appears to be the latest in a series of state-by-state confrontations over the issue.
"There have been some meetings with the governor's office indicating that they are receptive to some changes to the legislation," Skelos said in Albany, New York. "Those discussions are going to continue."
At last public count, 31 senators, including two Republicans, were in favor of the bill. Its backers need one more GOP member to vote in favor for it to pass.
Gov. Andrew Cuomo, who proposed the bill, says it would grant same-sex couples equal rights to marry "as well as hundreds of rights, benefits and protections that are currently limited to married couples of the opposite sex."
The first-term governor has lobbied opposition and undecided senators in an effort to secure the single vote needed for its passage.
Meanwhile, Greg Ball -- one of several Republican senators who are considered on the fence over the pending vote -- reiterated his party's concerns about the legislation's potential impact on religious institutions.
"I'm not going to vote for a bill where Catholic adoption agencies or religious organizations are shut down or are opened up to lawsuits," he said Friday, noting that adoption agencies were closed after same-sex marriage laws were adopted in other states. He declined to elaborate.
Democrat Tom Duane, an openly gay member of the state Senate, countered Friday that there are "already laws on the books in New York state which protect people of faith."
"Marriage is a civil right. People get married, and it is recognized in government," he said. "We're talking about the government recognizing same-sex equality."
Suzanne Goldberg, director of Columbia Law School's Center for Gender & Sexuality Law, called the potential for lawsuits a "red herring."
"New York state politics are at best unpredictable," she said Friday. "But this last-hour notion of marriage laws affecting religious organizations escapes the bigger issue.
"Religious institutions are generally either exempt or have already been operating against the state's existing anti-discrimination laws," she argued.
But those in opposition have voiced criticism that the measure would undermine traditional definitions of marriage, something that the bill's backers argue is long overdue.
Former Super Bowl icon and New York Giant David Tyree stirred controversy after suggesting Thursday that its passage would represent "the beginning of our country sliding toward ... anarchy."
But the passage of same-sex marriage laws faces a daunting history in the New York state legislature.
The Senate rejected a bill to legalize same-sex marriage in 2009 and has blocked the last three attempts by the state's lower house to get the proposed legislation signed into law.
Although New York currently does not grant same-sex marriages, a 2008 appellate court ruling upheld the right of couples to have their same-sex marriages recognized if they are performed elsewhere.
The current measure would amend the state's Domestic Relations Law to say, among other changes, that "no application for a marriage license shall be denied on the ground that the parties are of the same or a different sex."
Five states -- Massachusetts, Connecticut, Iowa, Vermont and New Hampshire -- and the District of Columbia currently grant same-sex marriage licenses.
CNN's Dana Garrett contributed to this report.