The 26-year-old has been a consistent performer for the Blues since his arrival from Marseille in 2012.
And he has gone on to make 171 appearances in his four seasons at Stamford Bridge.
But Spanish newspaper Marca suggest Bilbao are looking at Azpilicueta, who has two years left on his contract.
The report goes on to claim the defender does not have a release clause in his contract.
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Bilbao have a Basque player only policy but Azpilicueta was born in Pamplona, which is 117km away.
Last month, the full-back insisted he was motivated to correct the wrongs of the 2015/16 season under Antonio Conte.
He said: ”We want to be on the other side and we have to fight to be there next season.
“We know it is going to be tough but we have to show the spirit to do it again.
“When you win trophies and when you win games, it is easy to find the support but this season, even if we weren’t getting the results, if we only won a few games at home, they always supported us home and away.
“They were always behind us and this is the real support and we have to give back winning trophies as soon as possible.
“The next chance we have is next season and in a few months we will be back here trying to win trophies for them.”