The 25-year-old started multiple matches for France in North America on the back of another strong season for Roma, and Manchester United are understood to be leading the chase for his signature ahead of next season despite already strengthening their midfield this summer with the additions of Andrey Santos and Youri Tielemans.
Chelsea, Liverpool and Aston Villa have also been credited with interest.
According to Sky Sport, Saudi club Al-Hilal want to sign the Frenchman this summer and have already made contact with AS Roma over a potential deal. The Riyadh-based club are coached by Simone Inzaghi, who is likely to know Kone well from his time at Inter Milan, where the two would have crossed paths regularly in Serie A before Inzaghi's move to Saudi Arabia last year.
Despite the interest from the Gulf, reports suggest Kone still prefers a move to the Premier League for now, though he has not ruled out any particular league or destination as he weighs up his options for the season ahead. Kone is under contract at Roma until the summer of 2029, giving the Italian club a strong negotiating position even amid the growing list of suitors.
The midfielder made 29 appearances in Serie A last season, 28 of them starts, scoring twice and providing three assists as Roma pushed for European qualification. His combination of physical power and reading of the game has made him one of the more sought-after midfielders in Europe this summer, with his value having climbed further on the back of his performances at the World Cup.
Manchester United's interest comes despite the club having already committed significant funds to midfield reinforcements this window, suggesting Kone remains a priority target regardless of the business already done.
Roma, for their part, face a delicate balancing act between resisting offers for one of their most influential players and the financial realities of a club that has previously had to sell to fund its ambitions in Europe. With interest now coming from four countries, a resolution to Kone's future looks unlikely to be reached quickly.