Watch: 'Andy is cautiously optimistic'
Andy Murray is "cautiously optimistic" about returning to action "at some point this summer", says mother Judy.
The British three-time Grand Slam winner, 31, said in March that he was pain-free after hip surgery but his chances of playing singles at Wimbledon this year were "less than 50%".
Murray said the operation meant it was possible he would not be able to play professionally again.
"It is still early days so we will have to wait and see," Judy Murray said.
She told the BBC: "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 broke down in tears at the Australian Open in January, saying in his pre-tournament news conference that he planned to retire after this year's Wimbledon because of pain in his hip.
However, he said the first Grand Slam of 2019 could prove to be the last tournament of his career.
After a gutsy first-round five-set defeat by Spain's Roberto Bautista Agut, Murray appeared to soften his stance by telling the Melbourne crowd he hoped to see them again next year.
In his post-match news conference, he said he was considering the resurfacing operation primarily to improve his quality of life.
Murray had the hip resurfacing operation - which keeps more of the damaged bone than a hip replacement, smoothing the ball down and covering it with a metal cap - in London on 28 January.
American doubles player Bob Bryan had the same surgery last year and returned to action, alongside twin brother Mike, five months later. No tennis player has competed in singles after having the operation.
Murray 'pain-free' after hip surgery'Still a chance of Wimbledon?' - analysis
BBC Scotland tennis reporter Kheredine Idessane
There's no disguising the sense of quiet optimism emanating from the Murray camp at the moment.
The social media "thumbs up" from Andy Murray himself to his hip replacement; pictures of him enjoying a round of golf; his mum Judy now saying there's every chance he could be back on tour at some point this summer. Admittedly, that gives him plenty of wriggle room, as the summer tennis season drags well past September's US Open.
He won't be at the French at the end of May but is there a chance he could feature at some point on the grass in June? Queen's Club and Wimbledon would be the obvious targets, even if only on the doubles court.
However, if a pain-free, rested, rejuvenated Murray starts serious on-court weight-bearing work at some point next month, there is a possibility he will play singles at the All England Club in July.
He only gave himself a 50% chance of that a few weeks ago but it's certainly no less than that now. Quite a turnaround when you think that, in January, he was tearfully contemplating retirement.
