The Czech legend will retire at the end of the season, and the Gunners need a backup for Bernd Leno.
Arsenal will only have a summer transfer budget of £40million should they miss out on Champions League football next season, and they see Martinez as an option that will allow them spend elsewhere as manager Unai Emery plans to strengthen his squad.
The Argentine left for the Championship side in January on loan till the end of the campaign, and the north London outfit are monitoring him ahead of a potential return to the first-team for 2019-2020.
Martinez arrived Emirates from Independiente in 2010, but has spent more time away on loan than at the club.
The 26-year-old has had loan spells at Oxford United, Sheffield Wednesday, Rotherham United, Wolverhampton Wanderers and Getafe.
The Arsenal goalie has impressed since linking up with The Royals, conceding seven goals in the six games he has featured in, keeping two clean sheets and helping the Championship strugglers to a victory and four draws, and he seems to be doing enough to convince Emery of his quality.