Smith Rowe remains in discussions with Arsenal over a new contract, with his current deal expiring in 2023.
Villa were hoping to capitalise on any indecision by offering £25m for the 20-year-old.
But the Gunners quickly shut down advance and told Villa they would not accept an increased offer for their academy graduate.
That situation could change if Smith Rowe holds off on signing a new contract with Arsenal.
The north London club are determined to tie down the attacking midfielder, who joined them 11 years ago.