The England U-21 midfielder will be offered double his current salary, believed to be around £4k-a-week, in return for pledging his future to the club.
Milner is on course for a pay rise after triggering a clause in his contract by playing a certain number of games.
Kevin Blackwell has told chairman Gerald Krasner he wants to build the team around the youngster.
Spurs failed with a £4m bid last month and Manchester United are also keeping tabs on him.
Milner, 18, has said: "I’ve heard there’s interest in me and it shows you’re doing something right. But I’m happy at Leeds and I want to stay."