According to Sport (sport.es), the 30-year-old, who has shone for Denmark as the Danes have reached the last 16 at Euro 2020, could be surplus to requirements at Camp Nou, after the signings of Memphis Depay and Sergio Agüero.
West Ham, Brighton and Norwich have also been linked with Braithwaite, who can play anywhere across the front, including as a winger.
The forward spent 18 months in England, signing for Middlesbrough in 2017 from Toulouse for £9m, but he netted only nine goals in 40 appearances before joining Leganes in La Liga on loan, before a club record €5m move.
Then in February last year, granted an emergency exception to sign Braithwaite out of a transfer window due to injury, and triggered his €18m release clause, signing a four-and-a-half year contract.
He has eight goals in 53 Barcelona appearances, and though his release clause was set to €300m, Barca are believed to want much of the €18m they paid Leganes.
An unnamed Russian club is also thought to be interested.
Burnley have four senior striking options in Chris Wood, Ashley Barnes, Jay Rodriguez and Matej Vydra, but are looking for options out wide, having tempted Ashley Young, before the Inter wing back decided on a return to Aston Villa.