Braithwaite, 29, only joined Barcelona from Leganes in February when they were granted special permission to buy him.
And the Catalan club want to recoup as much of the €18m fee as possible by selling him to the Premier League.
Barca have good relations with Everton after selling Lucas Digne and Andre Gomes to the Merseyside club.
Meanwhile, West Ham have been interested in Danish international Braithwaite since his Leganes days.
However, they are not likely to pay the €18m that Barcelona want for Braithwaite.
Braithwaite has experience from playing in England after two seasons with Middlesbrough from 2017 to 2019.