Barcola, 23, has been linked with a move away from Paris Saint-Germain for several months, while the club had at the same time been keen to extend his contract.
A product of Lyon's academy, Barcola joined Paris Saint-Germain in 2023 for a reported €45m and has since won the Ligue 1 title in each of his three seasons at the club.
It had been unclear whether Paris Saint-Germain would be willing to sell the winger this summer.
French newspaper L'Equipe reported last week that the club are now open to letting Barcola go ahead of next season.
Talks over a new deal have been paused, and the two parties could part ways this summer if no agreement on an extension is reached.
Arsenal have been one of the clubs credited with interest in recent weeks, and that interest has now become more concrete.
Liverpool have also been linked with Barcola as they look to reshape their attack this summer, though reports suggest Arsenal currently lead the chase.
The Independent reports that the Premier League club are planning to submit a bid for Barcola in the coming weeks.
How much Arsenal are prepared to offer for the winger has not been disclosed, nor has it been made clear what fee Paris Saint-Germain would accept for the France international.
Barcola's contract with the French club runs until the summer of 2028.
He scored 11 goals and provided one assist in 29 Ligue 1 appearances for Paris Saint-Germain last season, and has already opened his account at the World Cup, scoring in France's win over Senegal.
Arsenal are looking to add further attacking depth to Mikel Arteta's squad as they bid to build on last season's Premier League title, with Barcola viewed as an upgrade on their current options out wide.