The Argentine star has largely been kept out by David De Gea at Old Trafford this season.
But Romero has been given added confidence following a run of games in the Europa League which culminated in him helping United beat Ajax in the final to secure Champions League qualification.
De Gea has been linked with a move to Real Madrid this summer and Mourinho would sign a big-name replacement if the Spaniard left.
But whatever happens, Romero is determined to remain at United and do all he can to become Mourinho’s first choice keeper.
He said: “I chose to come here because of what the club is. It’s a huge club – one of the biggest in the world – and one of the things I couldn’t leave alone. I couldn’t say no.
Sergio Romero insists he doesn't want to leave Manchester United
David De Gea has been strongly linked with a move to Real Madrid
“I knew it might be a little harder here to play a lot. Last year with Louis (van Gaal) I played 12 games and I felt good, so I thought I could be important for the team in those moments despite not playing a lot.
“Then with Jose I had a little more hope that I could play a little more. I’ve been able to show him I’m a goalkeeper that can play for this club.
“But you know never know what’s going to happen in the future, so you have to keep working hard for what comes next.
“I could be the No.1 or No.2, it all depends on hard work and the hard work I have. But my intention is to stay.”
Romero, meanwhile, hopes United’s Europa League triumph can somehow bring a bit of happiness to Manchester in the wake of the shocking terrorist attack that killed 22 people.
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He added: “We knew when it happened that it would be a nice prize for the people, although obviously winning doesn’t change anything for the families affected.
“There are families suffering who have lost a mother, father or child and football can’t change that at all. But all we can do is try and bring a bit of tranquility to the fans.”
Antonio Valencia has signed a new contract with Manchester United that will keep him at the club until June 2019, with the option to extend for a further year.
The Ecuador international captained United in their 2-0 win over Ajax in the Europa League final in Stockholm on Wednesday.
Antonio Valencia has signed a new contract with Manchester United
Valencia said: “Manchester United has been my life since 2009 and I am absolutely delighted to have signed a new contract.
“Wednesday night provided the one trophy this club had never won and it was a true honour for me to captain the team for the final.”
Mourinho said: “It is no secret that I had been an admirer of Antonio’s long before I joined the club.
“What I could never have imagined was what a great person he is. I truly believe it is a real privilege for us to have such a good player and such a good man.”