Midfielder Bakayoko arrived for a second spell at Milan on a two-year loan deal with option to buy for a reported 15 million euros ($17.7 million) after spending each of the last three seasons away from Chelsea.
Milan said they were delighted to welcome the marauding midfielder back and said they would be obliged to make the deal permanent if "certain conditions were met".
The Blues bought the 27-year-old in the summer of 2017 after he starred alongside Kylian Mbappe in Monaco's run to the semi-finals of the previous season's Champions League but struggled in the Premier League.
He was first sent to Milan on loan a year later when Gennaro Gattuso was coach, but the 'Rossoneri' did not take up their option to make the move permanent.
The following season he returned to Monaco on another year-long loan but again was sent back to Chelsea, before rejoining Gattuso at Napoli last campaign, playing 44 times as the southern side missed out on Champions League qualification on the final day.
With Gattuso gone and Luciano Spalletti in the dugout at the Stadio Diego Maradona, Bakayoko again went back to London this summer before making his return to Milan.
Bakayoko has played once for France, coming on as a substitute in a 2-0 friendly defeat to Spain.