Leonard forced his way out of San Antonio Spurs this offseason and penned a 12-months contract with the Raptors.
However, the short length of the deal has led to heavy speculation he could move onto pastures new next summer.
Various reports suggest Leonard is desperate to play in Los Angeles – either for the Lakers or Clippers.
But Webster is hoping the work his medical team have done in getting the small forward back to full health will help to convince him to stay in the Eastern Conference.
The Raptors are top of their division after 23 matches and are in with a strong chance of winning the Championship.
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“Yes [we are keen to keep Leonard],” Webster told TSN 1050.
“These are the internal talks and things that we have.
“I think his health has been better since day one, and that is important to him.
“It is all about being attentive and pushing him, all the things you would want to do with your own health.
“Our medical team are thinking for the long term and they are the best in the NBA. It is amazing what we can health and medical wise.
“And then what happens on the court by winning, him getting better and the team getting better.
“It is really about the team having a chance to make some serious noise.”
Webster also claimed that Leonard has not taken any notice in rumours linking him with a move away.
“Yes [Kawhi has told us what makes him happy] and from the very first moment in July and that’s obviously something that we keep confidential,” he added.
“As far as communication with him and understanding his need – yes.
“I think it has been documented that Kawhi is not a huge social media player.
“I would be surprised if he read a tonne of stuff.
“In that sense it is our job to monitor what is going on around the team and also not let it affect us.
“I do think Kawhi is unique in the sense of he is not every minute checking what is going on on the outside.”