Hatch, R-Utah, defended the federal Defense of Marriage Act, which prohibits the federal government from recognizing same-sex marriages, and hopes the court will allow each state to handle this controversial issue on its own.
"I'll say this, I do believe this could be solved greatly by a civil-union law that would give gay people the same rights as married people," said Hatch, who mentioned hospital visitations and tax benefits. "I think we can solve this problem without undermining the very basis of marital law in our country."
Six states offer civil unions, including Colorado and Hawaii, while nine others and the District of Columbia have legalized gay marriage. Same-sex marriage is illegal in the rest of the nation.
The senator and his office did not immediately respond to a request for comment and elaboration.
Hatch has previously supported constitutional amendments defining marriage as an opposite-sex union, and he touts that position on his Senate website, but in interviews during his 2012 re-election campaign and again Wednesday he said marriage should be up to the states.
Utah bans gay marriage, as did California, though its law is one that is being challenged in the U.S. Supreme Court. Hatch said it's quite possible the justices will reject the California case because of a lack of standing, a move that would result in gay marriage being legal in California. But he warned justices not to legalize gay marriage nationally, saying it would have long-term unintended consequences.
"If you start overturning things like that, I gotta tell you, there will be every Tom, Dick and Harry in there with some crazy marital scheme demanding the same rights and the same privileges."
Utah Democratic Chairman Jim Dabakis, who is gay, has strong disagreements with Hatch's position but sees his comments on civil unions as progress. "I'm really pleased that the senator is evolving," he said, using the same word President Barack Obama tapped in explaining his decision to start supporting gay marriage in early 2012.
"Having said that, I don't know why LGBT people in this country would settle for a second-rate kind of recognition."
Rep. Jason Chaffetz, R-Utah, said he is unsure of how such civil unions would work and added that with marriage, "The more government gets entangled with marriage the messier it becomes."
"I am open to looking at things, but I want to be crystal clear that I support traditional marriage," Chaffetz said. "Orrin Hatch is very conservative on this issue so if he has an idea, I'm all ears."
None of Utah's six members of Congress supports same-sex marriage.
In recent weeks, a number of Democratic senators and two Republican senators have announced they have switched positions and now support legalizing gay marriage. Only four Senate Democrats now oppose gay marriage.
Senate Majority Leader Harry Reid, D-Nev. and the highest-ranking Mormon in the government, announced his support for gay marriage last year, shortly after Obama made his own announcement.