The former Chelsea winger returned to the Premier League on loan from RB Leipzig in January to reinforce Ange Postecoglou's attacking ranks while Son competed at the Asian Cup.
Werner quickly earned Postecoglou's trust with two assists in his first two top-flight games for the club, while he netted in back-to-back wins over Crystal Palace and Aston Villa last month.
The Germany international made his seventh Premier League start for Tottenham in the midweek clash with West Ham United, setting up Brennan Johnson's opener before Kurt Zouma rescued a point for the Irons.
Prior to Tuesday's 1-1 draw with the Europa Conference League champions, Postecoglou was asked about the possibility of Tottenham triggering Werner's permanent option, believed to be approximately £15m.
Son: 'Werner is playing with more confidence'
The Australian refused to give too much away, only stating that the decision would be made at the appropriate time, but Son has now given his public backing to Werner sticking around in the English capital.
Reflecting on Werner's performance against West Ham, as quoted by the Evening Standard, Son said: "I think he listened to what the gaffer asked for. The way we approach the game, the wingers are very important, staying wide and being in the right position. And Timo and Brennan are doing a great job, both understanding each other.
"When he came to Tottenham, [Timo] was missing a little bit of confidence, but now you can see he's playing with confidence and more comfortable — missing a few chances, but that will come.
"He's doing a great job, but as I always said, he's a Golden Boot winner in the Bundesliga and we expect he'll score and give assists more for the team, because we have massive games ahead.
"I always want to make good friends and good team-mates, but in football there are no guarantees and the club will make a decision. But Timo would happily stay here, I can see that.
"I can say I want him to stay"
"As a player, I want to help him so he has a better option than right now. That's a team-mate's job. We'll see what's going to happen at the end of the season. I can say I want him to stay."
While Werner could still be donning the Tottenham colours for 2024-25, his prospective permanent signing could potentially force another left-wing option out of the door for good this summer.
Bryan Gil was viewed as a major coup when he joined the Lilywhites from Sevilla in 2021, but he has struggled to settle at Tottenham and has already been loaned back out to Spain twice, rejoining his former club for the second half of last season after a stint with Valencia.
Since Postecoglou took charge of Spurs, Gil has earned just 221 minutes across 11 appearances, making just two Premier League starts and being left out of the squad for their last five top-flight games.
The 23-year-old's Tottenham deal runs until the summer of 2026, but he is apparently eyeing a permanent return to Spain in the upcoming transfer window.