Frank Lampard is keen to bring in a new forward before the end of the transfer window as he looks to bolster his options.
Brereton could be seen as a replacement for Richarlison, who joined Tottenham in a £60million deal earlier this summer.
Lampard is also without Dominic Calvert-Lewin, who sat out of Everton's first two Premier League clashes through injury.
Anthony Gordon is also linked with a move to Chelsea, with the Toffees boss tasked with trying to keep hold of the 21-year-old.
According to Sky Sports, Ligue 1 side Nice had a £10million bid rejected by Blackburn for Diaz on Wednesday.
The 23-year-old was in prolific form last season netting 22 goals in the Championship and has attracted plenty of admirers.
Diaz, who can play on the left wing and through the middle, is open to a move.
The Chile international has scored two goals in his first three appearances for Blackburn this season, helping them reach the top of the Championship.
Everton do face a battle with Bournemouth for his signature, with Cherries boss Scott Parker also looking for more attacking options.
West Ham had also previously been credited with targeting the striker, but their interested has since cooled.
Should Everton fail in their attempts to sign Diaz, they could turn their attentions to other options.
Armando Broja, who netted 10 goals on-loan at Southampton last season from Chelsea, remains a key target and somebody Lampard knows well.
The Everton boss handed the Albanian star his senior debut at Stamford Bridge in March 2020.
However, with little progress in negotiations, they see Diaz as the perfect alternative.
Brighton's Neal Maupay and Coventry's Viktor Gyokeres are also players the Toffees are interested in.