The Italy U21 fullback joined the Bees on-loan in January to the end of the season, with the deal including a permanent option.
TMW says Kayode has done enough to convince Brentford to sign him outright.
An announcement could be made this week, though Brentford have now informed Fiorentina of their decision.
At the end of April, Andrea Ritorni, head scout of Vigo Global Sport who represents the defender, spoke about Kayode's farewell: "He is very happy to play in English football, he gets along very well with his teammates and the coach.
"He also played a couple of games with the club's Under 23s where he did really well. Changing clubs, environments, cultures is not easy and a few weeks of adjustment were also needed to get back into the right rhythm of play."