The Toffees are pursuing defensive duo Lucas Digne and Mina from Barca as Marco Silva looks to improve his back-line.
Mina is thought to have had talks with the Premier League club since the World Cup, where he scored three goals for Colombia.
It has been claimed that despite only recruiting the towering centre-back from Palmeiras in January, Barca would be willing to sanction a permanent departure with a buyback clause or a season-long loan with an up-front fee.
Leicester City, West Ham and even Wolves have been linked with the 23-year-old – who would likely cost over £25million if he were sold.
But though he faces a fight to break into Ernesto Valverde’s plans after a shaky first season at the Nou Camp, in which he made just six senior appearances, Mina says he wants to stay in Catalonia.
Speaking to Marca Claro: “I have a contract with Barcelona and I want to be successful there.
“There have been offers for me but right now I want to enjoy my holidays and then we’ll see.
“If they don’t want me, I’ll have to look for the best exit.
“But I remember that my intention is to triumph with Barcelona because it’s the best team in the world.”
Barca have added further competition for Mina in central defence after bringing in Sevilla star Clement Lenglet in a £35m deal earlier this summer.
Having also signed Arthur from Gremio, the La Liga champions are currently concentrating their efforts on Chelsea’s Willian – whom they recently bid £65m for.
Meanwhile, Everton are thought to be closing in on a deal worth up to £50m for Watford forward Richarlison to complete their first addition of the summer.
They also want Digne – who has a £90m release clause in his contract – from Barca having reportedly launched an £18m opening offer for the unfavoured Frenchman.