The 20-year-old, who has scored six Serie A goals for his Italian club this season, has been linked with the likes of Manchester United, Juventus, Arsenal and Barcelona over the last couple of months.
However, according to reports in France, Chelsea have attempted to steal a march on their rivals by submitting an offer of £21m ahead of the January transfer window, which opens next week.
Kessie started his professional career with Stella Club in Ivory Coast, before making the move to Atalanta in 2015. Last season, he scored four league goals during a loan spell with Cesena.
Earlier this month, it was claimed that Juventus had won the race for the Ivorian after watching him at close quarters this term.