The 51-year-old returns to the dugout nine months after he was sacked by Burnley.
Dyche will be officially confirmed as the new boss in the next 24 hours, according to the Daily Mail.
Ian Woan and Steve Stone will reportedly join Dyche as his assistants, having previously worked with him at Turf Moor.
Struggling Everton axed Frank Lampard on Monday with the club 19th in the Premier League.
Former Leeds manager Marcelo Bielsa had been owner Farhad Moshiri's first-choice for the Goodison hot seat.
He was in London yesterday for talks with the Toffees hierarchy, before reports claimed he had rejected the chance to become manager.
Sky Sports claim Dyche also held talks with Everton on Thursday.
Former Toffees boss Sam Allardyce was another considered, as was Davide Ancelotti, the son of ex-Goodison gaffer Carlo.
While former Southampton boss Ralph Hassenhuttl was also on the shortlist.
But ex-Burnley manager Dyche is the man tasked with leading Everton's fight against relegation.
He has been waiting for a top-flight offer after snubbing several approaches from Championship clubs.
Dyche will reunite with James Tarkowski, Michael Keane and Dwight McNeil at Everton, having worked with all three players at Burnley.
He takes over with the Toffees having won just three out of their 20 Prem games this season.
And Dyche will be thrown straight in at the deep end after penning a deal to take over.
His first game is a daunting meeting with league leaders Arsenal at Goodison Park on February 4th.
The Toffees then make the short trip across Stanley Park to face Liverpool at Anfield nine days later.
Dyche spent nine-and-a-half years with Burnley before being sacked by the Clarets last year.