The former Chelsea manager last week signed a contract until June 2019 with the record 20-times English champions.
"To speak of Jose Mourinho is to talk about a winning coach and that is United's philosophy and mine," De Gea told Marca on Friday.
"With that idea in mind, I'm certain we can build a great team. I welcome him."
Titles have been scarce for De Gea since the Spaniard moved to Manchester from Atletico Madrid in 2011.
He won the Premier League in 2012-13, Alex Ferguson's last season in charge of the team. David Moyes was then sacked 10 months after replacing Ferguson and his successor was Louis Van Gaal.
United finished fourth in the Premier League in the 2014-15 campaign and were eliminated from the Champions League in the group stages this season.
They were fifth in the league and must settle for the Europa League, but they did lift the FA Cup last month.
"When you speak of United nothing is enough if you don't win every title," De Gea said.
"Not qualifying to the Champions League was a big blow but we have won the FA Cup after 12 years, and after three years without lifting a trophy.
"We must value this because it's a very important trophy for the fans. This FA Cup should be the launchpad for the club to win important titles."
De Gea, 25, was close to leaving Manchester last year for Real Madrid but the clubs did not complete the deal before the transfer deadline.
He signed a new contract with United until 2019 but speculation about his future has continued.
"There's a lot of talk but little is known," De Gea said. "I understand how things work but what I feel is disrespectful is for someone to speak on my behalf. I have three further years left on my contract with United and I'm enthusiastic to win titles there."
De Gea is training with the Spain squad in preparation for Euro 2016.
(Reporting by Adriana Garcia, editing by Ed Osmond)
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