Webster's present deal expires in the summer of 2005 but The Jambos are eager to commence negotiations for prolonging the 20-year-old's stay at Tynecastle.
The former Arbroath stopper has been in sensational form this season and his performances have earned him a call-up into the senior Scotland squad.
Now, the SPL outfit want to reward Webster with a fresh contract and the player's agent revealed the defender would be happy to stay.
"Hearts want to speak to Andy about extending his contract and he will be delighted to sit down with them for discussions," said Charlie Duddy.
Such has been Webster's sparkling form, Scotland boss Berti Vogts has hailed the youngster as one of the best defensive talents in Europe.
Vogts said: "He is one of the best defenders I have seen of his age. I have watched a lot of good youngsters, particularly in Germany, but I would have to say Andy is up there with Europe's best 20-year-olds."
