The Spaniard has been heavily involved this season but has held a meeting with Perez and his agent.
The Real bigwig has given Asensio assurances over his position at the club and has agreed to not sign anyone that could harm his growth at the Bernabeu.
Spanish outlet Diario Gol say that means good news for Chelsea as the La Liga champions will not move for Eden Hazard.
However, they also claim the pact can be broken if Asensio’s form dips in the future.
The 21-year-old has played 1,112 minutes of football for Real this season, netting seven goals in the process.
Asensio impressed at the start of the season when deputising for the suspended Cristiano Ronaldo.
The reports claim Perez is amazed with the talent of the young forward and is desperate for him to stay long-term at the club.
Asensio will hope to help Real Madrid close the gap on La Liga leaders Barcelona who are eight points clear after 13 matches.
Chelsea ace Hazard has long been admired by Real with Zinedine Zidane a fan of his countryman.
The Blues star has been in good form this season, netting six times and providing six assists in all competitions.
Chelsea return to action on Wednesday evening when they welcome Swansea to Stamford Bridge.