Bryan, who returned to Grand Slam competition at the Australian Open following five months out with a similar injury, now has a metal hip implant after he sought surgery last year.
The American is the first tennis player to have undergone the operation, and it is an option Murray is considering. Former world No. 1 Murray said after his first-round Australian Open defeat to Roberto Bautista Agut that he would decide in the coming week whether to have further surgery or rest up until Wimbledon in July, which could be his last tournament if he decides to retire.
Murray has spent the last 18 months dealing with pain in his right hip and missed the majority of the 2018 season after he had his first surgery in January last year. However, the Scot still feels pain when playing and was visibly struggling with his hip during the five-set defeat to Bautista Agut on Monday.
"Seeing the way Andy Murray is feeling kind of hit a nerve with me," Bryan said at a news conference on Wednesday. "I would love to see him do a similar surgery, feel the relief that it gives. I think our hips are pretty similar: just worn down, no cartilage."
Bryan's recovery has provided hope to Murray, who has been asking the American how his hip has been holding up following matches and practices in order to gauge whether it would be a realistic option for him.
Bryan, 40, underwent the 90-minute operation in August before the US Open and had returned to full practise by the start of December. Bryan, who is back together with brother and playing partner Mike, said he has so far felt no negative effects and playing with the implant causes him "zero pain".
However, Bryan also said the recovery process for Murray would be different as the physical demand for a singles player is more intense than it is for a doubles player.
"I never once told him this is the way to go because I do see that singles is a different monster," Bryan said. "Those guys are really sliding around, killing themselves for four hours. Who knows if this joint would hold up.
"It's not going to break, but who knows if you have that little explosiveness needed to be super quick on the singles court. If you're a step slow, it's very exposed out there on a singles court.
"I'm just telling him, I feel great, quality of life is great, practices are going well. Maybe I'm not 100 percent yet, but I'm only five months. The doctors said this is more of like a seven or eight months until you feel perfect.
"Until I feel that, I can't give you the guarantee, but I think he's to the point where this is probably his last option."
Bryan has recommended a surgeon, Dr. Edwin Su, to Murray. Su, who is based at the Hospital for Special Surgery in New York, has helped athletes return to the NBA, MLB and NHL and has held initial discussions with Murray about the operation.