Gerrard will move to the Los Angeles Galaxy when his Liverpool contract expires at the end of the season.
The midfielder was courted in 2005 by Chelsea, only for a dramatic change of mind to see him stay at Liverpool, where he has played his entire club career to date.
Speaking after Chelsea’s 1-1 Capital One Cup first-leg draw at Anfield on Tuesday, Mourinho discussed the 34-year-old’s situation.
Mourinho was asked on BBC Radio 5 Live if he should have a word with Gerrard and persuade him to stay, and said: “Or to get him on loan.”
The tongue-in-cheek remark comes after long-serving former Chelsea midfielder Frank Lampard ended up with Manchester City this season, after an apparent deal with New York City FC seemed to mean his Premier League career was over.
Mourinho said of Gerrard: “I’m not happy he leaves. I would love the Premier League to keep always the best players.
“I think he has more than enough quality to play in the Premier League. In spite (of the fact) he’s my opponent, I feel sorry that he’s leaving.”