Sportsmail revealed on Monday that former Leeds boss Bielsa was the Toffees first choice candidate to replace Lampard, who lost his job following a dreadful run that has left the club in danger of relegation.
However, the Argentine is understood to have serious reservations about taking over at Goodison Park in a development that arrive as a major blow to Everton’s hopes of appointing Bielsa.
It remains to be seen whether the Merseyside club try to persuade Bielsa or move onto other targets.
Key figures within the Everton hierarchy remain admirers of their previous manager Sam Allardyce, who has held initial talks with the club over replacing Lampard.
There is a feeling at Goodison that Allardyce’s experience in guiding relegation threatened teams to safety offers the club with a strong chance of avoiding the drop.
Sean Dyche is out of work, but there is a reluctance from the Everton board to move for the ex-Burnley boss.