Turkish football pundit Ertem Sener said the Red Devils were expected to make the bid before the window closed at 6pm on Tuesday but it failed to materialise.
Serie A giants Juventus were also said to be considering making an offer for their 21-year-old playmaker, according to Radyospor.
According to the Daily Mirror, while the Gunners did make calls to sign frontman Cavani but decided against making an offer – because the 28-year-old has 'no sell-on value'.
Arsenal are the only side in the Premier League side not to sign an outfield player this summer, with Arsene Wenger's sole purchase being Petr Cech from Chelsea.
Chelsea new-boy Papy Djilobodji is setting his sights silverware after sealing a four-year deal at Stamford Bridge.
"I'm very happy to sign for a big club like Chelsea and I want to do my best to help the team win titles," he said.
"I want to thank the club for giving me this opportunity."
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Jose Enrique says he will battle for his place at Liverpool
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Meanwhile, Liverpool defender Jose Enrique has vowed to fight for his place at Anfield despite being linked with a late move to West Brom.
"Just to let you all know that it wasn't any offer from West Brom for me," Enrique wrote on Twitter.