The north London club are looking to bolster their attacking options after Harry Kane was ruled out until March with a hamstring injury.
Jose Mourinho has no other natural centre-forwards at his disposal, and will therefore delve into the January transfer market to find a solution.
The Guardian reports that Spurs will provide competition to Villa and Newcastle for Poland international Piatek, who could be on the way out of San Siro.
Piatek has struggled in his first full season with the Rossoneri, after finishing last season strongly following his January move from Genoa.
The Pole has scored just four goals in 18 Serie A appearances and Milan are willing to listen to offers, especially now that Zlatan Ibrahimovic has re-joined the club.
However, Spurs and others are likely to have to fork out the same £30 million fee that Milan paid for Piatek a year ago to secure his services.
All three Premier League clubs are understood to prefer a loan move, but the Serie A side are after a permanent sale.
Spurs could also be restricted by a lack of funds, but the sale of Christian Eriksen, who is out of contract in the summer, could aid their bid to sign a new striker.
Inter Milan are weighing up a £20m move for the Denmark international.
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