The Reds, Barcelona and Aston Villa had been linked with the 21-year-old winger with Unai Emery's side previously seeing a reported bid rejected for the £42million-rated player.
Part of the hope behind the interest was the fact that Williams' contract was due to expire at the end of the season, however he has now agreed to an extension to 2027 with Bilbao.
"I am happy in Bilbao," he was quoted as saying on the weekend about his future. "People have to stay calm, I have to keep working."
Williams has continued to impress for Athletic this season, racking up seven assists as he nears a century of games for the LaLiga team despite his tender age.
— Athletic Club (@AthleticClub) December 1, 2023
Liverpool's January transfer plans are eagerly awaited by the club's fans as they put together a challenge for the Premier League title.
The Reds are also into the Europa League last-16 after smashing LASK 4-0 on Thursday night.
After the game, Jurgen Klopp praised young centre-back Jarell Quansah for justifying his decision not to sign a new centre-back.
"There couldn’t be better news for the club, to be honest," the manager told reporters.
"Before this season, people told us we should buy a centre-half. We knew we had Jarell. Did we know that he would show up like this? Not exactly, but we were pretty hopeful. So the future’s bright."