The Hammers are planning for next season having secured their Premier League status and are believed to have offered Ambrosini a two-year deal.
The 35-year-old, a multiple UEFA Champions League and Serie A winner, is out of contract next month and had initially planned to see out his career at the San Siro.
However, he is bellieved to be tempted by the prospect of ending his 18-year stay at Milan to try his hand in the Premier League.
Roggi said: "There is interest from West Ham and it's a real interest, like that of other English, German and American teams, although there have not been concrete contacts.
"But, like every year, the first thing Massimo will do at the end of the season is speak with Adriano Galliani (AC Milan CEO)."