Matri will fill the hole left by injuries to Giuseppe Rossi, leading scorer in Serie A with 14 goals, and German international Mario Gomez.
"There are some minor details to eke out and then I will be available for the coach," Matri told ViolaChannel.tv.
"I am hungry to play and to score a lot of goals," added the 29-year-old, who has played just six times for Milan this term, scoring once, and has become the target of abuse from the fams.
Matri will sign a six month contract with no option for Fiorentina to buy him.
Fiorentina have been searching for a striker since Rossi was injured on 5 January while Gomez has been out of action since virtually the start of the season.
According to the Italian press Fiorentina are also on the verge of signing on loan Brazilian forward Anderson from English champions Manchester United.