An initial proposal to take the 25-year old on loan is understood to have been rejected.
The Poland international is also wanted on loan by Newcastle and Aston Villa but Milan have been holding out for £30m.
Piatek wants to leave following the arrival of Zlatan Ibrahimovic and an injury to Harry Kane means Spurs are set to accelerate their interest in Piatek.
The 24-year-old could be available this month provided Milan's valuation is met.
Tottenham had preferred to wait until the summer to sign a striker to rival Kane - but their vice-captain's injury has changed their approach to the January window.