Williams was close to joining Barcelona last summer, but resisted their interest and penned a new deal with Athletic in August.
However, the Spain star could still leave San Mames at the end of the season, with the Gunners pushing for a deal.
Former Premier League scout Mick Brown, speaking with Football Insider, stated: "Arsenal have been working on getting a deal done for a long time. He will be their priority this summer.
"Their new sporting director (Andrea Berta) will benefit from his contacts here. But I think it will be a difficult deal to get done regardless of which club it is.
"I think Real Madrid have a good chance. I don't know of many players who have turned down Real when given the opportunity to go there. But if Arsenal could get into the driver's seat, talk to his agent and so on, then yes, then we could very well come to an agreement."