Sky Deutschland have today reported that Gbamin, 23, could sign a five-year contract with the Blues later this week, with a medical possibly taking place on Thursday.
The deal is expected to cost Everton about £27.5 million, after an initial bid of £23 million was rejected by the Bundesliga club.
The Ivorian would be a timely arrival for the Toffees after fellow defensive midfielder Idrissa Gueye left Everton for Paris Saint-Germain for £30 million on Tuesday after three years at Goodison Park.
Gbamin, who has four years left on his current deal at Mainz, has made 95 appearances for the German side since joining from Lens in July 2016, scoring three times.
He did not feature in Mainz’s 3-1 friendly win over Everton on Saturday, having represented his country in the African Cup of Nations earlier this month.
Meanwhile, Everton seem extremely close to another recruit, having agreed a fee with Juventus for striker Moise Kean on Tuesday.
Manager Marco Silva has said he is targeting five more signings before the transfer window shuts on Thursday, August 8, and Gbamin and Kean would go some way to filling the most obvious voids in Everton’s squad.