The 20-year-old impressed during his first season in the Premier League, making 36 top-flight appearances, but the right-back could not prevent the Canaries from being relegated to the Championship.
Aarons only signed a new long-term deal with Daniel Farke's side last summer and has a contract with the second-tier side until the end of the 2023-24 campaign.
According to The Athletic, Barca lodged two separate bids for Aarons, one of which could have been worth £20m plus bonuses, but Norwich rejected the approaches.
Aarons played the full 90 minutes of Norwich's 2020-21 Championship opener at Huddersfield Town last weekend. body check tags ::