Kaka has been linked with a move away from San Siro after it was revealed there was a clause in the Brazilian's contract allowing him to leave if the club did not qualify for next season's UEFA Champions League.
Milan are in 11th position in Serie A, 22 points adrift of the final Champions League spot.
The 31-year-old, who is out of contract in 2015, has made no secret of his desire to ply his trade in the MLS one day.
But Galliani is determined to see Kaka remain in Italy after he celebrated his 300th appearance for Milan with a brace against Chievo last weekend.
"The hope of the club, and of Ricky too I hope, is that he stays and celebrates his 400th, 500th and who knows how many more appearances," Galliani told Milan Channel.
"He is in the hearts of all Milan fans. He is still a kid, he can still play in so many other matches.
"I am happy to have met him and I hope he is happy to have found Milan. I hope he can stay with us for a long time."
Kaka has been a key performer in a disappointing Milan side following his return from Spanish giants Real Madrid in September.
The Brazil international has scored nine goals in all competitions for Clarence Seedorf's team.
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