Milan have indicated they are only willing to spend around £13m on the right-back but Spurs value him at more than £22m.
Tottenham boss Jose Mourinho is willing to let Aurier leave this summer but Spurs want roughly what they paid Paris Saint-Germain for the Ivorian three years ago.
Mourinho is targeting a replacement right-back for Aurier and has been linked with Lille’s Zeki Celik.
Milan are trying to push down the price for Aurier and are considering PSV Eindhoven right-back Denzel Dumfries as a cheaper alternative.
Spurs have completed the £15m signing of Pierre-Emile Hojbjerg, with Kyle Walker-Peters heading the other way to Southampton for £12m. Both players have signed five-year contracts. It means Spurs are effectively paying £3m, plus Walker-Peters, for Hojbjerg.
Hojbjerg becomes Mourinho’s first summer signing. After securing a sixth-placed finish and a Europa League spot next season, Mourinho is looking to strengthen further this summer.