A poll suggests that just over half of Minnesota voters do not want a constitutional ban on same-sex marriage.
The state may hold a referendum on the issue next year if the House follows the Senate in approving a bill.
The Star Tribune Minnesota poll found that 55 per cent of respondents did not want a ban, compared to 39 per cent who said they did.
The newspaper said this was a “sharp reversal” of poll results taken seven years ago.
Minnesota already bans gay marriage but opponents say that the law must be strengthened to ward off challenges by gay rights campaigners.
Although support for gay marriage is growing across the US, voters given the chance of a referendum on the issue have always come out against marriage equality.
Of the 31 states which have voted on constitutional gay marriage bans, all have approved the measures.
Speaking to the Star Tribune, gay Democrat Senator Mark Dibble said: “Public opinion has been moving on this issue for some time, probably farther and faster than most folks expected, including myself.”
But he added: “We also know people are going to be asked this question after the end of an 18-month campaign, in which millions of dollars are going to be poured in to influence their opinion and the outcome.”