However, he had an inauspicious start, with Gattuso bemoaning Bakayoko's "defects" in the aftermath of his debut against Napoli.
A continued run of substitute appearances followed but the Frenchman has been a mainstay since breaking into the Milan side for October's 2-1 win over Genoa.
This week, Bakayoko told Corriere dello Sport that it is his "dream" to remain with the Rossoneri and Gattuso is enthused by such a prospect.
"At this moment we cannot do without him, he must remain at Milan," he told a news conference ahead of Sunday's trip to Roma.
"He is proving technical skills. You know that I am not in charge of the finances, but he is certainly giving us fundamental improvements.
"He can become an important player for this club for many years."
Another Gattuso recruit, Krzysztof Piatek, enjoyed a dream full debut in midweek as he scored both goals in the 2-0 Coppa Italia quarter-final win over Napoli.
Milan supporters are excited by the prospect of Piatek being paired with Patrick Cutrone in attack, while his coach is keen to leave an instinctive finisher to his own devices.
"I do not like the word 'managed', your instinct must be left alone," Gattuso explained.
"We must not give him too much information, we must leave him free to follow his instinct.
"They [Piatek and Cutrone] can play together, it's true. But then we have to see if we want to play with a three-man midfield, with the playmaker and how many players are left out.
"I do not like to leave Cutrone out too much."
Milan are a point ahead of their opponents in fourth heading into the weekend and Gattuso added: "In times of difficulty they have always done great performances. They have great players and a coach who has been doing well for years. We must be very careful."