The striker's current deal with the Santiago Bernabeu side is due to expire in 2015, but they are eager to hold on to him beyond the expiry date of his existing contractReal Madrid are ready to open contract talks with Alvaro Morata and offer the Spain Under-21 international a new and improved deal.
The promising attacker's existing deal is due to expire in the summer of 2015 and several clubs are believed to be closely monitoring his situation at Madrid.
Inter president Erick Thohir recently admitted that he would like to bring Morata to Serie A, while the striker has also been linked with clubs such as Juventus, Napoli, Borussia Dortmund and Arsenal.
Nevertheless, Madrid see Morata as an important player for the future and will offer him a fresh deal at the end of the 2013-14 campaign.
"Morata has been doing a great job this season. I hope he will stay at Madrid. We need him. He has the potential to be close to Karim Benzema and Cristiano Ronaldo," head coach Carlo Ancelotti said last month.
There's no guarantee Morata will continue his career at the Santiago Bernabeu side, though, as he could be tempted to move on in order to get regular first team football elsewhere. The 21-year-old has netted six goals in 19 La Liga appearances this season, but has mainly been warming the bench behind first choice striker Benzema.
Madrid hope to reach an agreement with Morata by the end of the month, but could make the striker available for around €12 million (£9.8m) if he opts against a contract renewal.