Sky Italia said that the Serie A club had offered £6m plus £1.5m in add-ons for the Gunners midfielder.
Wilshere has 12 months left on his current Arsenal contract, but spent the whole of last season on loan at Premier League rivals Bournemouth.
The 25-year old was a fixture in the middle for Eddie Howe’s side at the Vitality Stadium until his season ended early when he broke his leg against Tottenham in April.
Wilshere, who has 34 England caps, has been linked with a loan move to Crystal Palace, and now Sampdoria, who finished 10th in Serie A last season.
The news comes just after Arsenal boss Arsene Wenger said that he expects Wilshere to remain at Arsenal and fight for his place.
"I think Jack will be with Arsenal next season, yes," Wenger said at the weekend.
"I will give you a very focused answer on that. He has to work hard to come back and fight for a place in the team.
"But once he is ready, when he comes back, he will not be far from practising with the team."