Nerazzurri boss Antonio Conte is in the market for a new left-back and has already been linked with a move for Chelsea star Alonso, who he worked with during his time with the Premier League side.
Alonso is valued in the region of £30million and Inter would have to pay that to have any chance of doing a deal after a loan switch was ruled out by the Blues.
Now reports claim Inter would also be interested in Alonso's team-mate Palmieri and are ready to sound out Chelsea to see which of the two is available.
Chelsea themselves are in the hunt for a new left-back and are open to letting one of them go so they can bring in a new one of their own.