The England international, whose current contract with Spurs expires in 2019, has been in North London since 2007 and is eager to spend the remainder of his playing days at White Hart Lane just like current club ambassador King did during his career.
"I'm 26 now and starting to think about if I want to finish my career here and I'd love to do that," Rose told the club's official website.
"There won't be many players able to say that they've been at a club for x number of years.
"Ledley is a legend. When you see him with the players, some will ask 'what was he like?' and there really are no words to describe how Ledley was as a player and as a man.
"He was quiet, but a leader. I saw the pain he had to go through to be out there on a Saturday. You wouldn't believe he'd missed most of the training sessions throughout the week and yet he'd be the best player on the pitch.
"I'd love to be able to look back at the end of my career and think I'd been at a club for the whole of my career. That's the plan.
"You don't know what will happen in football but that's the plan, I want to be here for a long, long time and I hope while I'm here I get to win trophies as well."
Since arriving at Tottenham from Leeds United almost 10 years ago, England international Rose has spent time on loan with Watford, Peterborough United, Bristol City and Sunderland, but is now a permanent member of Mauricio Pochettino's line-up.