Carrick, 35, signed a one-year deal yesterday which runs through until June 2017 - his 11th season since joining from Tottenham in 2006.
Mourinho, who was appointed as Manchester United manager on May 27, wants the correct blend of youth and experience within his squad.
And his eagerness to find that balance led Mourinho to sanction a contract extension and even made it one of his top priorities, according to The Times.
After seeing his veteran enforcer sign the deal, Mourinho told ManUnited.com: "Michael is a very intelligent midfielder and a great reader of the game.
"I am pleased that he will be extending his contract. His form this year shows his ability and his enjoyment of the game are as strong as ever.
"Michael has a wealth of experience from his many years at the club and that knowledge will be invaluable to me. I am really looking forward to working with him."
Carrick added: "This great club has been a part of my life for the past 10 years, so I am delighted that this incredible journey is continuing.
"It's great to have the opportunity to work under Jose Mourinho, who has achieved so much during his managerial career.
"I would like to thank the fans for their unwavering support. Winning the FA Cup was a special moment and, hopefully, we can go on to win more trophies in the future."
