If he makes it he will join stars such as Sir Chris Hoy, Katherine Grainger, Glasgow-born Michael Jamieson, and Neil Fachie in a procession from Kelvingrove Art Gallery and Museum, through the city centre to George Square.
Murray won a gold medal at the London Olympics just a few weeks ago.
After his victory in New York on Monday night he said: "It's a shame I missed the parade in London. I would like to try to go back for the one in Glasgow. It would be nice.
"I'd like to go home to see my family and just be around them. I obviously haven't got to see them that much and they've always supported me.
"I will make time to go back to see them but I don't know whether it will be this weekend or not.
"But I would also like to get a few days to myself just to take everything in and just be normal for a few days before doing any of that.
Asked if he was prepared for the reception when he got home, Murray said: "I don't know if you can necessarily prepare for it.
"I think being around the Olympics and seeing how the nation came together, everyone was just right and it was a nice thing to be involved in."