The 18-year-old joined North London outfit Tottenham from Milton Keynes Dons on a five-and-a-half-year contract on transfer deadline day.
Tottenham paid MK Dons £5 million for the services of the midfielder, according to The Sun.
Former Spurs midfielder Anderton believes that Alli can be a success at White Hart Lane, but has urged him to keep his feet on the ground.
“The last thing he must do is think that because Spurs have paid five million and given him such a long contract that he has made it as a player,” the 42-year-old told The Sun.
“We’ve seen in the past — with players like John Bostock — that what looked an exciting move does not work out.
"So my biggest advice to Dele would be to focus purely on becoming a better player and listen to the coaching he will be getting at Tottenham.”
Alli is rated as one of the best young players in the lower divisions in English football, and over the past two years he has indeed been superb.
The midfielder has skills, technique and an eye for goal, and does look to be a good signing for Tottenham in the long term.
Anderton’s advice to Alli is indeed very sound, and one which the youngster should take on board.
Perhaps the fact that the teenager will continue to play for MK Dons on loan for the rest of the season will see him remain humble.