Having beaten Argentina's Leonardo in four sets on Friday, on Sunday the defending champion meets German world number 47 Florian for a place in the fourth round.
Murray has met the 29-year-old twice before, both times on clay.
He has come out on top each time but both matches were close, with the most recent coming this year in Madrid, when Murray got the better of two tie-breaks.
Mayer, who has an unorthodox double-handed slice backhand, is a two-time Wimbledon quarter-finalist and a member of the exclusive club of players not to have a losing record against Rafael Nadal - they have split their two matches.
Murray said: "He's very tough. He comes to the net a bit. He has a good slice backhand, but he hits it with two hands, which isn't how normally you're taught to play that shot.
"He has very good feel up at the net, he hits a lot of drop shots. He's a pretty flat ball striker. He has sort of huge, looping strokes. It can be tough to time your split step and know when he's going to make contact with the ball.
"He's caused a lot of guys problems over his career. I'll need to play well to beat him."
Murray will hope he does not have as difficult a time as he did against Leonardo Mayer on his least favourite show court in grand slam tennis.
The Scot enjoys the atmosphere on Louis Armstrong but seems incapable of playing his best tennis on what is a very tight arena.
All his recent US Open struggles against lesser opponents have come on Armstrong, including his last early loss at a grand slam against Stanislas Wawrinka in the third round in 2010.
Murray had been clinical in beating Michael Llodra in round one, but he is not concerned about his dip in form.
He said: "I thought the first match was very good. I didn't feel like I served as well as I did in the first match. It was a different court, different conditions during the day.
"I finished the match well. I played well when I needed to. That's a good sign.
"Obviously I want to keep improving as the tournament goes on. You don't want to play your best right at the beginning, so I hope I can get better."
Florian Mayer beat America's Donald Young in round two and will take confidence from his close shave in Madrid into tomorrow's match.
The German said of the win: "I'm so proud of myself, only the victory mattered. So often I failed in the past when I was the favourite and supposed to win, so I'm really looking forward to the match against Andy.
"I can relax and enjoy it without any pressure - and if I go on playing like this, maybe it's time to cause a little sensation. I have nothing to lose and my match against him in Madrid was great. If I'd played better on the set points, I could have won the match."
Murray will certainly hope to walk out on Ashe, which would mean a return to the biggest stadium in tennis for Mayer nine years after he first played there.
"In 2004 I played on Arthur Ashe against Andre Agassi and it was an incredible match," said the German. "Unfortunately I got injured and had to retire, but I won one set and it was great."