The 24-year-old has been heavily linked with a move away from San Siro as he has less than eight months remaining on his contract.
According to El Nacional, PSG and Barca are preparing to table contract offers for Kessie and will negotiate a deal in January, ahead of a proposed free transfer for the Ivorian next summer.
The report adds that PSG would seemingly be the favourite of the two clubs to land the midfielder as the Ligue 1 outfit would be able to offer him an "astronomical contract" with a "very high salary", one the Catalan giants may struggle to match given their current financial predicament.
Kessie has been a mainstay in Milan's midfield so far this campaign, helping the club claim 31 points from a possible 33 available which sees them sitting joint top of Serie A along with Napoli. body check tags ::