The former Chelsea and Everton hit man has hinted himself that he could seek to leave Old Trafford and would like to play in Italy.
The French giants are keen for the mis-firing Belgium international to replace Edinson Cavani and would be willing to give United back most of the £90 million they paid for him.
But they face strong competition from Inter who have been given Lukaku’s name on a wish list of players by Antonio Conte who is close to taking over at the club next season.
Conte wanted to take Lukaku back to Stamford Bridge when he was in charge of Chelsea and is keen for them to link up in Milan.
Lukaku has previously made clear he would one day love to play in Serie A.
“Playing in Serie A is a dream, it would be really a dream,” he explained recently.
“I hope to be able to play sooner or later even if at the moment I am focused on United.”
Reports emerged earlier this week that Solskjaer would hold talks with Lukaku over his future at Old Trafford.
Since the Norwegian has taken over from Jose Mourinho, the former Everton striker has fallen down the pecking order behind Marcus Rashford.
And Lukaku could consider his future this summer as a result.
The Red Devils would likely demand a big fee from either PSG or Inter Milan should they make their interest concrete.
Lukaku could feature for United in their final game of the season on Sunday against Cardiff in the Premier League.
Whether it turns out to be his final game for the club remains to be seen.
Lukaku still has three years left to run on his contract at United.