Spurs had verbally agreed on terms with Fonseca over a two-year deal, but negotiations between the two parties broke down on Thursday.
Sky in Italy claims the deal was scuppered by financial reasons.
Spurs’ new managing director of football, Fabio Paratici, is a huge fan of Gattuso and considers him a more appealing appointment.
Gattuso left Fiorentina today, 23 days after having joined as manager.
This follows a dispute with the Serie A club over which players to sign this summer.
The club has now terminated his contract.
Paratici had tried to appoint former Inter Milan head coach Antonio Conte but talks broke down earlier in June.
