The Ivorian has been linked with both clubs this summer as they seek reinforcements.
According to the Daily Star, Seri has a £35 million release clause in his contract that is set to expire later this summer.
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Chelsea are now in a predicament with their delay in recruiting Maurizio Sarri as their next head coach but they remain optimistic that a deal could be struck before they face Perth Glory in their opening pre-season game.
Seri has developed in to one of the best midfielders in French football since joining Nice from Pacos de Ferreira in 2015 and he was virtually ever-present last season as he provided 2 goals and 10 assists to help Nice finish 8th in the Ligue 1 table.