Stones pulled his hamstring in the first half of City's comfortable 2-0 win over Leicester City on Saturday and looks set for a prolonged spell on the sidelines.
Kompany, who has endured a torrid time with injuries over the past few seasons, made his latest comeback in the game at the King Power Stadium after almost three months out and put in a commanding display.
The Belgium international believes he has learned enough from his own injury experiences to know that if you put the work in, you can come back stronger — advice he is keen to share with Stones.
"Stonesy was one of the outstanding performers so far," Kompany told reporters after the game.
"If there's one thing I can give him it's that it's something you have to go through in your career at times and there's no reason why you shouldn't come out stronger for it.
Great team performance at a difficult ground today, it was a great feeling to be back with the lads on the pitch. Thanks to the fans again for your support, we’re all wishing John a speedy recovery! #ManCity pic.twitter.com/cnNVvJyWCJ— Vincent Kompany (@VincentKompany) November 18, 2017
"Hopefully it's not too bad and when he comes back we'll need him as much as we'll need anyone — otherwise we won't achieve all these targets we have."
Kompany escaped with a yellow card after hacking down Jamie Vardy in the third minute of a game that made it 10 league wins in a row for Pep Guardiola's side.
The City skipper was adamant his challenge was no more than a bookable offence.
Kompany said: "I knew it was a yellow, it was early doors, it was a good run from Vardy and I thought, 'Just use my joker now'.
"But then afterwards as well, what was positive was that I still managed to play virtually the entire 90 minutes with a yellow card in the back and still do what was needed for the team, put in some good challenges, put in some good tackles.
"So overall I was happy with my performance, especially after coming back."