The 20-year-old is one of the rising stars in European football following a string of impressive displays for Atalanta BC in the Italian top flight.
Kessie has featured 21 times for Atalanta this term in all competitions and also starred for Ivory Coast at the Africa Cup of Nations, playing in all three of their group-stage outings.
Reports in January suggested that Chelsea had a £27m bid knocked back for the youngster, while Man United are also said to have shown an interest following his recent admission that he 'dreams' of playing for them in the Premier League.
According to the Daily Star, Kessie's representatives met with Roma last week and a verbal agreement was reached, but there is still hope of an English club coming in for him and handing over the full £34m asking price.
Kessie, who joined Atalanta from Ivorian side Stella Club in 2015, is also understood to have been pursued by Everton in the past 12 months.