The Argentina goalkeeper is expected to leave Villa Park this summer.
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Martinez has been linked with Manchester United and Barcelona in recent weeks. However, The Sun says a return to Arsenal could also be on.
The Gunners see Martinez as a potential replacement for David Raya should the Spain international be tempted away.
Raya is seen at Real Madrid as a replacement for Andriy Lunin and also is on the radar of Atletico Madrid.
For his part, 32 year-old Martinez would love a return to London Colney, particularly if offered No1 status.