Leno is yet to recover and Martinez kept the shirt as Arsenal beat Liverpool on penalties in the Community Shield on Saturday.
The 27-year-old has made it clear he is unwilling to stay in the Emirates to replace Leno, believing he has done enough to prove himself.
And Leeds are monitoring the situation with Marcelo Bielsa ready to sell Kiko Casilla, the Daily Mail reports.
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The Whites have already signed Ilian Meslier for a permanent deal, but could also move for Martinez if they can sell Casilla.
Former Real Madrid goalkeeper Casilla has been erratic between the sticks at times at Elland Road and was banned for eight games last season after being convicted by the FA of abusing Jonathan Leko.
Arsenal initially wanted £ 20million fees for Martinez, Valencia and Real Betis among other interested clubs.
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But they could be forced to lower their asking price if Arteta decides Leno is his No.1 in the future.
Leeds have been busy in the transfer market since returning to the Premier League for the first time since 2004.
They have already added Valencia forward Rodrigo in a deal worth up to £ 30million, and have forked out £ 13million for German international defender Robin Kock from Freiburg.