The 20-year-old impressed for the Gunners during the 2020-21 campaign, contributing four goals and seven assists in 33 appearances in all competitions.
Smith Rowe's current deal at the Emirates Stadium has two years left to run, but the club are thought to be in talks over a new long-term contract in North London.
According to Goal, Villa approached Arsenal with a £25m bid for the youngster but were immediately told that he was not for sale this summer under any circumstances.
The attacker, who joined Arsenal's youth system in 2010, finished the 2020-21 Premier League season with two goals and four assists in 20 appearances. body check tags ::