The 32-year-old has so far been restricted to doubles action since returning to the game following his hip resurfacing operation in January, winning his first tournament back at Queen's in June before suffering early exits at Eastbourne and Wimbledon.
He will link up with brother Jamie at this week's Citi Open in Washington.
Speaking ahead of the tournament of his return to singles action, he told reporters: "I'm closer than maybe what I thought I was."
Murray added that a return to the hard courts in August was in his sights, and if not, then he would wait until after the US Open.
He said: "Best-case scenario probably would be Cincinnati. And then if I wasn't able to play in Cincinnati, there's a good chance I would probably wait until after New York, because I wouldn't want my first tournament, either, to be playing best-of-five."
Murray's last grand slam appearance was on the hard courts of Melbourne, where he was dispatched in the first round by quarter-finalist Roberto Bautista Agut in four sets.
The Murray brothers last played together at the Rio Olympics, but were a prolific partnership during Great Britain's 2015 Davis Cup victory, while they have won two ATP Tour titles together, in Tokyo in 2011 and Valencia the year before.
Jamie is the defending champion at Washington, having won the tournament last year with Bruno Soares.