Unai Emery is a big admirer of Williams and has been keen to take him to Villa Park this summer.
The Premier League side put a lucrative offer to Williams to switch to England, something Athletic Bilbao could not compete with.
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However, according to Spanish radio programme El Partidazo de COPE, Williams has rebuffed the interest from Aston Villa and will stay at Athletic Bilbao.
The attacker will now extend a contract at Athletic Bilbao which is due to expire in 2024, as he intends to continue in La Liga.
Williams grabbed six goals in La Liga last season, providing five assists for his team-mates.
The Spaniard also struck on three occasions in the Copa del Rey as Athletic Bilbao reached the semi-final stage.
Athletic Bilbao will now be quickly looking to lock Williams down to a fresh contract to further strengthen their position on the winger.