The Premier League duo are said by a report in El Mundo Deportivo to want to sign the Denmark international, who is unable to play for the Blaugrana again this season due to not being registered for the Champions League.
Real Valladolid, Sevilla and Real Sociedad are also keen on the striker, who scored just one goal for the Catalan giants across 403 minutes of first-team action.
It is claimed the club would be keen to recoup the €18m they spent on him, with Everton holding a particularly good relationship with the Blaugrana board following deals for Lucas Digne and Andre Gomes.
Ousmane Dembele and Luis Suarez have both returned from injury while Lionel Messi is the star forward with Antoine Griezmann and Ansu Fati both ahead of Braithwaite in the pecking order.
The Denmark international had netted six goals and provided one assist for struggling Lega across 24 La Liga appearances this campaign before moving to the Camp Nou in February.
Formerly of Middlesbrough, Braithwaite has penned a four-and-a-half year contract at Barcelona in a deal which will contain a €300m release clause after the club were granted an emergency transfer by the authorities.
This ruling provoked some controversy, as Lega were powerless to stop Braithwaite leaving – due to his release clause being activated – but were then unable to sign a replacement themselves.