The Sun says Jackson has rejected approaches from Serie A pair AC Milan and Napoli as he wants to stay in the Premier League.
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United are interested and have been informed by Chelsea intermediaries that Jackson can leave for £80m.
Jackson will be allowed to go by Chelsea management after the arrival of fellow strikers Joao Pedro and Liam Delap since the end of last season.
Jackson remains tied to Chelsea until 2032, though is falling out of the plans of manager Enzo Maresca after being sent off twice in his last four games.
Along with United, also watching developments are Aston Villa, where manager Unai Emery worked with the Senegal international at Villarreal.