King hoping for Kane stay at Tottenham amid Bayern Munich bid
Kane, whose 30 Premier League goals last season were not enough to help Spurs seal a European place, has been the subject of two bids from Bayern.
Both of those have been rejected by Spurs, though with Kane out of contract next year, there remains a possibility the England captain might finally end his long association with the club.
King, though, is hoping Kane will stay and play a key part in Ange Postecoglou's team.
He told Stats Perform: "It's always important to keep your best players, of course.
"Harry's an unbelievable player and what goes on from here is obviously up to him and the club. But him being here, being as professional as possible, working as hard as possible is what we're used to.
"That's Harry in a nutshell. I know he's enjoying the training, he's enjoying performing under Ange at the moment, and the rest we leave between him and the club to sort."
While Kane's future remains in doubt, one player to come through the door at Spurs this transfer window is James Maddison.
The playmaker scored 10 goals and provided nine assists in 30 league appearances for Leicester City last season, though could not save the Foxes from relegation, but King thinks the Maddison is the ideal fit for Tottenham's midfield.
He added: "I feel like he's the type of player that we need in the squad at the moment. He is a creative player. He's a player that can shoot from outside the box, technically, very good, and wants to receive the ball in tight areas.
"He's a really good fit. He had a very good season last season with Leicester and he's an England international. These are the type of players that we want to attract."
While King has been "very impressed" by Postecoglou's start at Spurs, he believes it is too early to put firm targets on next season.
"First and foremost, I think we want to see the players have an identity in terms of the way we play and I think Ange brings that," King said.
"He has a certain style and the way he sees the game, which I think will suit our players, suit our club and excite the fans.
"In the Premier League, you don't know what it's going to throw at you. You start the first game with the intention to try and win every game, and you take it from there.
"The top four, of course, would be fantastic. That's where we'd love to be. But we realise how difficult that is.
"Just to have a season where players perform at their highest level, we play good style, a brand of football, and something that we can work on will be good."