Lawmakers say they expect the state Senate to take up the law Tuesday and the state House to do the same soon after.
House Speaker Salvatore F. DiMasi and Senate President Therese Murray back the repeal, but their support alone may not be enough to secure other lawmakers' votes, The Boston Globe reported Thursday.
Supporters of gay marriage rights say they are optimistic the repeal will be approved because of backing from leading lawmakers and Gov. Deval Patrick.
"If we get the 1913 law repealed, it brings us one very important step closer to full equality," said Arline Isaacson, cochairwoman of the Massachusetts Gay and Lesbian Political Caucus.
Opponents argue the law gives out-of-state couples an unfair advantage.
"I just feel that it would be hypocritical if I turned around and said, 'Fine, you can come here and get married, and we'll recognize it'," state Rep. James R. Miceli, D-Wilmington, told the newspaper.