Everton have had an offer accepted by Crystal Palace for the winger, which could rise as high as £30-million (R540 million), according to SkySports.
The 27-year-old would become the most expensive player in the Merseyside club’s history, an honour currently held by Romelu Lukaku, who they signed from Chelsea for £28-million (R504-million) in 2014.
Everton’s coffers have been bolstered on Tuesday by the sale of defender John Stones to Manchester City for a fee believed to be in the region of £47.5million (R855-million).
They are also expected to announce the signing of Swansea City defender Ashley Williams in the coming days.
Bolasie has spent four years at Palace, scoring 13 goals in 143 appearances.