Moyes agrees to make Everton return: reports
The club's new owners, the Friedkin Group, moved for Moyes after dismissing Dyche just hours before Thursday's 2-0 win against Peterborough in the FA Cup third-round.
Moyes is reported to have agreed a two-and-a-half year contract to return to Everton 12 years after leaving for an unsuccessful spell in charge of Manchester United.
The 61-year-old Scot has been out of work since leaving West Ham at the end of last season.
He is set to start his second spell as Everton boss against Aston Villa in the Premier League on Wednesday.
Moyes remains popular with Everton fans after his 11-year stint in charge from 2002 to 2013, a period that included a fourth place finish in 2005 and an FA Cup final appearance in 2009.
Dyche's two-year reign came to end on Thursday after Everton's defeat at Bournemouth on January 4.
Former Chelsea boss Graham Potter was said to be Everton's top target but he joined West Ham, with Jose Mourinho, currently in charge of Fenerbahce in Turkey, also reportedly distancing himself from the job.
Moyes' appointment is expected to be confirmed on Saturday and he will be tasked with leading Everton away from the relegation zone as they prepare to usher in a new era at their Bramley Moore Dock Stadium from next season.
Dyche managed only one win in his last 11 games in charge, leaving them just one point above the relegation zone.
Former Everton defender Leighton Baines, now the club's under-18s head coach, and captain Seamus Coleman took charge for the win against third tier Peterborough.