Corriere dello Sport says Liverpool have outlined their contract offering to Kessie.
The Reds continue to be linked with the Ivory Coast international, despite his public desire to remain with Milan.
Liverpool are prepared to offer Kessie a contract worth £115,000 per week.
Kessie's current deal at Milan expires next summer, and the player has publicly stated his wish to remain at the club for the foreseeable future.
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