Mina's stock soared during Colombia's run in the World Cup in which he was responsible for three goals, allowing Barcelona to make over three times what they paid.
Andre Gomes, meanwhile, will sign on a season-long loan - with Everton paying a loan fee of £2m for the Portuguese worldwide midfielder.
BBC reports that Mina's arrival means Everton have spent more than £60m in the past two weeks following the initial £35m arrival of Brazilian winger Richarlison from Watford, a deal that could rise to £50m.
Bernard began his professional career at Atletico Mineiro in 2010 before joining the Ukrainian champions three years later, where he won the Ukrainian Premier League, Ukrainian Super Cup and Ukrainian Cup three times each.More news: Deaths reported as massive quake hits Lombok
More news: Yemen: Dozens of children killed in airstrike
More news: Israeli fire kills 2 Hamas militants in Gaza
'I know about Everton's players and I look forward to being able to play with my new team.
The free signing of Brazilian midfielder Bernard was confirmed after a work permit was granted. I have spoken a lot to the coach and I know his passion for Everton and determination to be successful.
Another addition from the Nou Camp, midfielder Andre Gomes, joined on a season's loan.
He said: "I chose Everton because of all the things I had heard about Marco Silva and after speaking to him I was happy with what he said to me".
Brighton and Hove Albion secured defensive reinforcement in the shape of former Barcelona right-back Martin Montoya, who joins from Valencia on a four-year deal.