Kane suffered a "significant lateral ligament" injury to his left ankle during Spurs' 1-0 Champions League win over Manchester City on Tuesday and is unlikely to play again this season.
The England captain is susceptible to ankle damage -- this is his fifth such injury since 2016 and the second to his left joint this year.
Surgeon Mark Davies, a leading specialist at the London Foot and Ankle Centre, who has operated on Premier League footballers, said going under the knife could help restore the strength of Kane's ligament to what it was before his first injury.
"At some point I think they would think about doing something surgically to stabilise the ligament, which is quite feasible and should restore the problem happening in the long term," Davies told Britain's Press Association.
"He's not that old, he could do with a stable ankle if he wants to carry on playing long term without it happening again," he added. "It is a routine operation because the ligament needs to be tightened up and that is fairly easy.
"If he were to have surgery, the surgeon would almost certainly use an internal brace, which is a device which you put over the ligament repair and it is incredibly strong.
"It means you are not just relying on scar tissue regaining strength so it would improve his chances of coming back."
The surgery would put Kane out of action for about three months, which may not be much longer than he could be expected to be out anyway.
The England striker has previously shown impressive powers of recovery and is targeting a return to fitness for a potential Champions League final and UEFA Nations League semi-final in June.
Davies, who has operated on Spurs, Arsenal and Chelsea players in the past, said Kane's quick recovery rates have not made him more susceptible, neither has a full-on schedule which has barely seen him get any rest over the past few summers.
"There is no merit in him having an extended rest, they are not sending him back too early, it's just that he has got an unstable ankle," Davies added.