The 26-year-old was behind Oleksandr Zinchenko in the Gunners' pecking order last season.
Tierney was made to settle for a place on the Arsenal bench for the majority of the campaign, and could leave the Emirates in search of regular game time.
The full-back has been linked with Newcastle and Manchester City, but he is said to not be ruling out a move back north of the border.
According to The Sun, the Scottish Champions are running the rule over him, and keen to re-sign him.
The reports adds a season-long loan is an option, but Tierney would have to take a pay-cut on his current £120,000-a-week salary.
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Tierney joined Celtic as a seven-year-old in 2005 and spent five years in the first team, before leaving for the Premier League giants.
He went on to make 170 top flight appearances, securing the Scottish Premiership and domestic cup four times.
The full-back would go on to sign for Arsenal in a £25m deal in 2019, and has won the Community Shield and FA Cup.
Newcastle were linked with a £30m swoop back in April, with Eddie Howe a huge fan.
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Pep Guardiola has also been credited with an interest, but it remain to be seen whether City swoop.
Either way, a move away from north London would suit both parties, and Celtic would offer the chance for Tierney to resurrect his career.