The Blues have held a long-standing interest in the winger who enjoyed a magnificent season with Athletic last term, racking up eight goals and 19 assists in 37 appearances.
Williams, however, signed a new long-term contract with the Basque club in December last year which contains a release clause of around £49 million.
And various reports suggest that the fee is too much for Chelsea who are looking for cheaper options to boost their attacking depth out wide.
While Chelsea may have pulled out of the race, Premier League rivals Arsenal are still interested in pursuing a deal this summer.
Mikel Arteta sees the player as ‘one of his preferred options’ in attack but the club also hold reservations over the asking price and hefty salary likely required to lure Williams from his boyhood club.
Williams, meanwhile, distanced himself from rumours of a possible move last week and insisted he was ‘happy’ with Athletic Bilbao.
‘I’ve seen reports talking about my future, but honestly, I have no clue of anything,’ he said.
‘Right now, I want to focus on the Euros and nothing else.
‘I am so happy in Bilbao, I signed a new deal and performed well last season. I feel at home, but I don’t know what I will do. Nobody can tell you about the future.’
For Chelsea, it marks the latest blow in their attempts to bolster their squad ahead of Enzo Maresca’s first season in charge.
The club have signed centre-back Tosin Adarabioyo from Fulham but missed on the coveted signature of Crystal Palace winger Michael Olise who is now reportedly on the verge of a £60million switch to Bundesliga giants Bayern Munich.
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