Reds manager Jurgen Klopp is in the market for a new midfield addition after losing Gini Wijnaldum this summer.
Sportmediaset says the Reds have intimated to Kessie's minders that they're willing to offer a contract worth €8m-a-year.
The Ivory Coast international is currently in Japan for the Olympics, but is being kept up to date on developments.
For his part, Kessie's preference is to stay with Milan, where he is seeking a new contract worth €6m-a-year. His contract runs to June 2022.