Williams, who is Barcelona's leading transfer target this summer, has been heavily linked with a switch to Hansi Flick's side over the last few weeks.
Arsenal and Liverpool are among the Premier League teams to be linked with the forward, but it is understood that the Catalan giants have always been at the head of the queue for his signature.
According to El Chiringuito TV, Barcelona have now come to a full agreement with the agent of the Spain international over a long-term contract at Camp Nou.
Journalist Jose Alvarez claims that Williams has been offered a five-year contract with increasing wages each season.
Williams 'closing in' on move to Barcelona
Barcelona are allegedly now waiting for Williams to give the green light, with the club then planning to officially activate his release clause.
Athletic have always maintained that the forward will not be allowed to leave for under his buy-out fee, which is believed to be €58m (£48.7m).
Despite their well-documented financial problems, Barcelona are said to be confident of completing a move for Williams, and registering him with La Liga, without any issues.
The forward was vital to Spain's Euro 2024 triumph, scoring twice and registering one assist in six appearances at the competition, including a goal against England in the final.
What will Williams bring to Barcelona?
Williams entered Euro 2024 off the back of an excellent season for Athletic, scoring eight goals and registering 19 assists in 37 appearances in all competitions.
The forward was then one of the stars of the European Championship, with his pace, clever movement and impressive end-product causing a whole host of problems to the opposition.
Williams impressed alongside Barcelona's Lamine Yamal in the final third, and it is now looking likely that the pair will also be teammates at club level.
In the current market, €58m (£48.7m) for a player of Williams' ilk is a bargain, and it would be the statement signing that Barcelona need in a summer which has seen Real Madrid sign Kylian Mbappe.
Once Williams has given the official go-ahead, the deal is set to advance quickly due to the presence of a release clause in his Athletic contract, so an announcement is expected to arrive in the near future.