Margaret Court, a 24-time Grand Slam singles champion and now a Christian pastor, is strictly against gay marriage.
Australian former world number four Sam Stosur hinted earlier on Tuesday (May 30) that players may refuse to play in protest.
Murray said it would be best to resolve the issue before the tournament begins but added: "We'll see what happens."
The Scot beat Russia's Andrey Kuznetsov in four sets to reach the French Open second round on Tuesday.
When asked if he would support players refusing to compete, the world number one replied: "I think if something was to be done it would be a lot more beneficial to do it before the tournament starts.
"For players to be in a position where you're in a slam and kind of boycotting playing on the court, that would potentially cause a lot of issues.
"If something was going to be happening and the players come to an agreement, if they think the name should be changed or whatever, that should be decided before the event starts.
"But I would imagine a lot of the players would be pretty offended by that."
Court, 74, has said she will not fly on Qantas "where possible" in protest of its support of same-sex marriage.
"I think everyone can have their opinion. I don't agree with it, but I guess we'll cross that bridge when we get to the Australian Open," Stosur, 33, said earlier on Tuesday.
"We'll see who wants to play on Margaret Court Arena and who doesn't."
The Australian number one added: "I find it very hard to believe that the name would ever be changed - the court's named Margaret Court Arena because of what she did in tennis."
Tennis Australia have said they will not rename the Arena, stating that Court's personal views are her own.
The venue was originally called Show Court One when it opened in 1988 before it was renamed in 2003 in tribute to the multiple Grand Slam winner.
Stosur's fellow Australian player Casey Dellacqua, who has two children with partner Amanda Judd, tweeted her opposition to Court: "Margaret. Enough is enough".
And Stosur issued her support for friend Dellacqua, saying: "It's been pretty fiery. Casey was obviously very adamant, and I wanted to support my friend and that's why I sent out my first tweet in a very long time."