The France international joined Milan on loan at the start of the campaign, with the Italian club also having an option to buy the player for a €35 million (£30.7m/$40.1m) fee.
Bakayoko has enjoyed his time in Serie A so far after struggling to make an impact at Stamford Bridge, making 25 appearances in all competitions for the Rossoneri despite a poor start to life in Italy.
He has grown to be a key figure for Gennaro Gattuso's side, with Milan currently fourth in the Serie A table and on course for a return to the Champions League next season.
Qualification for Europe's premier club competition would likely lead to Bakayoko being able to remain with the seven-time continental champions, and he is hopeful that his future will lie at San Siro.
"It's my dream [to stay], the thought fixed in my mind," he told Corriere dello Sport. "But I have a contract with Chelsea.
"[Milan chief] Leonardo says the deal is tied to the Champions League. I'm not in a position to talk about it. I just want to finish the season in fourth place, then we'll think about it."
Bakayoko is also appreciative of Gattuso's coaching style after the Milan boss helped him settle in at the club.
"Gattuso was not my ideal player, although I say that with great respect because he won the Champions League, which is my dream," he said. "Please do not tell him...
"For me, he is the boss and a nice person who has helped me a lot. In the first two or three months, when everything was not going well, we talked every day.
"Now we understand each other with a glance and our relationship is perfect. He has a talent to show love to his players."
Milan are next in action on Sunday when they visit Stadio Olimpico to take on fifth-placed Roma in a key clash in the battle for a top-four finish.