The Italy international could struggle for playing time next season, as he sits below Antonio Valencia in the United pecking order.
Mourinho has said that Darmian can leave the club if they receive a suitable bid.
Laurentiis has said that Napoli have made a loan offer with a view to a permanent deal.
Speaking to Italian 'Radio Kiss Kiss', Di Laurentiis said: "We have asked for Darmian on loan with a right to sign next summer."
Darmian joined United from Torino in 2015, and according to reports, sees a move back to Italy as an attractive prospect.
"I want to play regularly," the 28-year-old said. "Last season, I didn't play much, that's my objective and my ambition. That's why I spoke with the manager and also with the club about my future."
"I want to leave, but we will see what happens because we don't know."