The Portuguese boss is out of contract with the Ukrainian side at the end of the season and was considered by the Blues when they were searching for a replacement for Ronald Koeman.
Fonseca reportedly has a wish to manage in the Premier League and would be open to relocating to Merseyside. The 44-year-old is said to be a part of Farhad Moshiri’s summer shake-up with Director of Football Steve Walsh also reportedly on the chopping block.
The Shakhtar boss has led his side to the Champions League knockout stages thanks to wins against Napoli and Manchester City and was said to be an impressive target for Moshiri and the Everton hierarchy when considering a replacement for Koeman following a setback in their hunt for Marco Silva.
According to Daily Mail reports, Allardyce’s future on Merseyside depends on his side's end to the season with a top-seven finish still within reach despite a disastrous season for the club as a whole.
This could be it for Allar, Allar, Allardyce
Although Allardyce is under contract through next season, his time at Everton is seen as a marriage of convenience between both parties by many, with the Blues needing the 63-year-olds guidance to remain in the Premier League.
The former England boss has reportedly begun planning for next season, however, he is still an unpopular figure with the Goodison Park faithful with many fans accepting that he will be the man in charge until the end of the campaign and not longer.