The Sun says Italian midfielder Gattuso, 31, is said to be disappointed at Ancelotti's exit from AC Milan and is considering Stamford Bridge.
Gattuso's agent Andrea D'Amico said: "The rapport between Gattuso and Ancelotti is very strong.
"He was deeply distressed by the decision to change the boss. Gattuso is very attached to Ancelotti the manager.
"Gennaro has the characteristics and the qualities needed to play in the English league."