United are looking to take advantage of the winger's stalled new contract talks with Napoli.
Sportmediaset says United are offering a five-year contract worth €8m net per season plus one in easily achievable bonuses.
Napoli would like to renew the agreement with the Georgian, but president Aurelio De Laurentiis does not intend to go beyond €6m-a-year, bonuses included. If the parties were not to reach an agreement, the Neapolitan club would consider the idea of selling him, but for no less than €100m.
PSG are also seriously interested, while Barcelona is more on the sidelines.
Last week, Napoli sporting director Giovanni Manna told TMW: "We want to reward his journey in Napoli. He still has two more years with us. We need to find an agreement on every point.
"We've already talked about it and if it doesn't work out, we'll talk about it again at the end of the season. We agree with the player not to distract us from the pitch."