The Sun says the club's captain and longest-serving player wants a long term deal - believed to be three years - when his current one expires at the end of the season.
But at 32, Chelsea are unlikely to offer anything more than the standard one year extension to older players.
There is a hope that loyal servant Azpilicueta could swing it with an option for a further 12 months beyond 2023.
The Spanish defender has been at Chelsea since 2012 and has won every honour there is at club level since then.
He is also a renowned fitness fanatic with an impeccable injury record who has barely missed a training session, let alone a match, in those nine years.
Even so, Chelsea have a tradition of only offering short term contracts to players over 30 regardless of their history with the club.
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