Sam Allardyce is currently the heavy favourite to get the job at Goodison Park, but O Jogo appear to believe that Fonseca is the preferred candidate.
Fonseca is currently in charge at Shaktar Donetsk in Ukraine, and he has done an impressive job.
The Portugues boss has also taken charge of the likes of Porto and Braga during his managerial career, but he has never before had control of an English side.
The 44-year-old’s appointment would be a huge risk for Everton, given their current predicament.
Everton are sat in 16th place in the Premier League table, and they simply have to get their next appointment right.
There are growing fears that Everton could be relegated this season, with their form going from bad to worse.
Allardyce has never before suffered a relegation as a manager, and he is also very experienced in Premier League management.
However, Fonseca would arguably be a more forward-thinking appointment, and the Portuguese press seemingly believe that he could be in charge at Goodison Park in the very near future, after claiming that the manager's talks with the Merseyside club are now in their final stages.