According to Sky Sports News, the Hoops are exploring a pre-contract and loan offer for the 27-year-old.
Arsenal will not trigger the Scotland international's one-year contract extension - as reported by The Athletic - and his current deal is due to expire in June.
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Tierney is now free to sign a pre-contract agreement with non-English clubs from January 1 and could secure a permanent transfer as soon as next month.
A number of clubs in England and across Europe are also said to be interested.
Tierney joined the Gunners in 2019 for a fee of around £25 million and spent the 2023/24 season on loan at Real Sociedad due to limited first-team opportunities at The Emirates.
The full-back has just come back from six months on the sidelines after suffering a hamstring injury in Scotland’s 1-1 draw with Switzerland at Euro 2024 in June.
He returned to action for Arsenal in Wednesday night’s EFL Cup quarter-final win over Crystal Palace.
Brendan Rodgers could look to strengthen the left-back area in the January window given the uncertainty of Greg Taylor and Alex Valle's futures.
Taylor's deal is set to expire in the summer, while Valle is on a season-long loan from Barcelona.