Carrick, whose one-year deal expires in June a month before his 36th birthday, revealed last week that this could be final campaign at Old Trafford.
But Mourinho, who believes Carrick brings the best out of record buy Paul Pogba, wants to keep the midfielder for another year.
“Michael is 35 years old and it is very sad to know that time flies for every one of us,” he said.
“He is such a fantastic player and I’ve always loved him.
"It is just a pity instead of being his manager when he was 25, I am his manager when he is 35.
“But I am lucky I’ve still got Michael Carrick – and probably for one more season.”
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The former England international has been a mainstay in the Manchester United side since signing from Tottenham for £18million in August 2006.
The veteran has amassed a huge collection of honours during his time at Old Trafford - including five Premier League titles.
Carrick was also part of the United team that won the Champions League in 2008, scoring in the penalty shootout victory over Chelsea in Moscow.