Gareth Bale has given a strong indication that he expects to remain with Real Madrid next season.
Bournemouth have signed Artur Boruc on a permanent deal following his release by Southampton.
Striker Morata has been heavily linked with a January loan move to Arsenal and Tottenham but remains confident he has a part to play in Madrid's Primera Division title push under coach Carlo Ancelotti.
He said: "I feel that the boss really trusts me, so does (Ancelotti's assistant Zinedine) Zidane, all of the coaching staff and the club .
"I am 21 years old, I am happy at Real Madrid and I want to be here for a long time.
"A player always wants to play and the manager has shown that he trusts in me. He has played me in a lot of difficult matches and I am very happy. I have to repay his trust in me. "
After a relatively slow start which saw them fall behind Barcelona and Atletico Madrid, Madrid have been impressive in recent weeks, notching five successive wins in which they have scored 23 goals to move three points behind their title rivals.
"The team is phenomenal. I hope we can carry on like this for a long time," said Morata.
"If we carry on like this we will be one of the most dangerous teams in Europe. Anybody who plays against Madrid already knows this."
Madrid take a break from league action this weekend with a Copa del Rey clash at third-tier Olimpic Xativa.
Morata may get the nod ahead of Karim Benzema and is hoping Los Merengues can go one better than last season's final defeat to Atletico.
"We are very enthusiastic. We know that it will be a very big match for both clubs," he said on the club's official website.
"They will be very excited to be playing against Madrid and we, knowing what the atmosphere will be like there, are very eager to play and to enjoy the atmosphere.
"You have to approach the match without disrespecting anyone and knowing that any match is important. It is a trophy and we want to win everything."