Wenger has seen his side's title challenge fall away in recent weeks, and the Gunners are now in danger of dropping out of the top four in the Premier League.
They lead Everton by a point, but the Toffees have a game in hand over their Champions League-chasing rivals.
Arsenal play Wigan Athletic this weekend in the FA Cup semi-final, and Savage feels that the Frenchman must step down as manager if they fail to lift the trophy next month.
He told The Mirror: "Arsene Wenger will run out of excuses if Arsenal don't win the FA Cup, and so fail to win a trophy for a ninth season in a row.
"Defeat against Wigan in the semi-final at Wembley on Saturday will surely signal the end for Wenger - especially if his Gunners also finish outside the top four and miss out on Champions League football next year.
"At any other big club, going a whole decade without putting a cup in the trophy cabinet would not be tolerated. And as much as I regard him as a legend - one of the all-time greats of English football - Wenger cannot trade on past glories for ever."
Arsenal last won a trophy when they emerged victorious on penalties in the 2005 FA Cup final against Manchester United.