AC Milan has apologised for a joke made by the club's sporting director Massimiliano Mirabelli regarding the signing of striker Nikola Kalinic from Fiorentina.

Kalinic completed his loan move to San Siro on Wednesday (AEST) and Milan has an obligation to complete a permanent deal for the Croatia international.

During a live-stream of Kalinic signing his Milan deal, Mirabelli said: "Now enough jokes, put the ball where you know, otherwise we'll send you back there [to Fiorentina]."

But Fiorentina reacted angrily to Mirabelli's comment, releasing a statement that criticised the Milan director for a remark the club described as "disparaging, out of place and entirely avoidable".

"We would like to remind Mr Mirabelli that Fiorentina is an important club whose players have made history in the game over the years," Fiorentina's statement read.

"Moreover, this club represents Florence, one of the most beautiful and well-known cities in the world. The club therefore asks that it be afforded the respect it deserves from today onwards."

After the release of the Fiorentina statement, Milan issued a swift apology from Mirabelli regarding his Kalinic joke.

"I have always had great respect and admiration for Fiorentina," Mirabelli said. "If someone perceived a lack of respect in my words, I apologise.

"I think it was obvious, by the tone, my expression, the body language, that what I said was a colloquial joke, certainly not with denigratory tone."

Kalinic is the 11th signing of a remarkable transfer window for Milan, with the forward joining the likes of Andre Silva and Leonardo Bonucci at San Siro.