The Scot is due to make his competitive return from hip surgery in doubles at next week's Queen's Club grasscourt tournament, partnering Spaniard Feliciano Lopez.
The U.S. Open starts on Aug. 26.
"I hope at some stage this year I would be able to get back to playing singles again," the three-times Grand Slam champion told British media, describing his current situation as "baby steps".
"I'm not really interested in putting a time limit on it because I'm quite happy just now so I don't need to play singles after Wimbledon or the U.S. Open.
"If I can, that would be brilliant, but I don't think that's going to be the case. I think it's going to take a bit longer," he added.
Murray, 32, has not played since losing in the first round of the Australian Open in January, after which he had a resurfacing operation on his right hip.
He said he was "pretty much pain free" now and enjoying training, including some singles practice sessions.
"I don't think when Wimbledon finishes that I will just step onto the singles court the following week and everything's good," he said. "I still have quite a lot of work to do before I'm at a level where I feel like I'll be able to be competitive."
(Reporting by Alan Baldwin, editing by Christian Radnedge)
