Australian Court, a 24-time Grand Slam singles champion now a Christian pastor, has long opposed gay marriage.
"If something was to be done, I think it would be a lot more beneficial to do it before the tournament starts", Murray said.
In the Margaret Court Arena's tweet they state, "Melbourne & Olympic Park does not support Margaret Court's comments and we remain an organisation committed to embracing equality, diversity and inclusion; from our fans to our colleagues who delivers the events that people love to attend".
The legendary Margaret Court's public opposition to same-sex marriage has sparked furious debate around whether the show court named in her honour at Melbourne Park should be changed.
The Court controversy flourished with contributions from tennis stars Casey Dellacqua, who told Court "enough is enough", and Sam Stosur, who hinted at a player boycott of Margaret Court Arena at next year's Australian Open.
It caused backlash from the likes of Aussie tennis player Casey Dellaqua, who tweeted "Margaret, enough is enough" and retired tennis player Martina Navratilova, who said "Thank you Qantas for your support". "If it's two men, two women, that's great". Your statement leaves me no option but to use other airlines where possible for my extensive travelling', Court said.
In an interview with Sky News on Monday evening, Court said her views had not changed.
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Naratilova was supported by fellow tennis great Billie Jean King, who is also gay, after both had been critical of Mrs Court's views on homosexuality in the past.
Australia has been criticised by worldwide human rights groups over its slowness to act on same-sex marriage.
Stosur, the 2011 US Open victor, doesn't believe Margaret Court Arena would change its name, but said a player boycott could be a reality next year.
Ms Court stood by her views in a live interview on The Project on Friday.
This time, she doubled down on her criticism of gay marriage, telling Andrew Bolt that "God made man for woman and woman for man ..."
"But I would imagine a lot of the players would be pretty offended".
"I'm all for any type of relationships".
"But it was still definitely a very quick turnaround given the length and the toughness of the match I played against Dasha [Gavrilova] just on Saturday".