Murray posted an Instagram video of himself on an outdoor court earlier this month, describing the footage of him hitting a tennis ball for the first time since the surgery as "a start".
He followed that up with a video of himself playing a round of golf that reinforced the message that he was on the mend.
"It's still early days so we will have to wait and see what happens. He is cautiously optimistic about getting back on the match court perhaps at some point over the summer," Judy was quoted as saying by British media.
"He was told not to do impact work, which basically means running around the garden hitting a ball, for three months but he's been hitting against a wall from a static position."
Murray hoped to make his comeback at Wimbledon and said in March that he would consider playing doubles because it would put less stress on his hip.
The 31-year-old twice Wimbledon champion has compared his situation to that of American doubles player Bob Bryan, who returned to competitive tennis 5-1/2 months after undergoing a similar procedure.
(Reporting by Shrivathsa Sridhar in Bengaluru, Editing by Simon Jennings)
