Arsenal have been offered the chance to sign Alphonse Areola on a season-long loan from PSG, according to reports.
Mikel Arteta is facing a selection headache ahead of the new Premier League season, which begins this weekend.
Bernd Leno was Arsenal's first-choice goalkeeper for the majority of last term, before injury cut short his campaign in June.
Having been handed a taste of regular action at Premier League level, Martinez does not want to go back to sitting on the bench.
The Argentine, who has been at Arsenal since 2010, may therefore seek a transfer if Arteta decides that Leno is his go-to shot-stopper for 2020/21.
Villa's second bid reportedly comes close to matching Arsenal's £20m valuation of the 28-year-old.
The Gunners will only sanction Martinez's departure if they are able to sign a replacement, so Areola's availability would be a boost to Villa.
The Paris-born goalkeeper is out of favour at the Parc des Princes, having recently returned from a season-long loan spell at Real Madrid.
Thomas Tuchel does not consider Areola to be in his plans this season, and Arteta has been alerted to his availability.
However, Arsenal could face competition for the Frenchman's signature, while Areola himself might prefer a move to a club that can guarantee him game time.
The Gunners begin the new season with a trip to Fulham – who have also been linked with Areola – on Saturday.
Their first home game of the 2020/21 campaign will come against West Ham the following weekend.
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