The Uruguay international has emerged as the Arsenal manager’s top target ahead of the summer transfer window, according to Spanish publication El Gol Digital.
Gimenez, 23, has re-established himself as a regular in Diego Simeone’s side in recent months after previously falling out of favour.
Atletico Madrid bosses would likely hold out for a hefty transfer fee but his contract contains a £57.3million (€65m) release clause.
El Gol Digital claim Arsenal have turned their attention to Gimenez after admitting defeat to Liverpool in the hunt for Lazio’s Stefan de Vrij.
Holland international De Vrij will leave the Serie A side when his contract expires at the end of the season.
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Lazio sporting director Igli Tare revealed last week that they would “say goodbye” to the central defender after failing to agree terms on a new deal.
Arsenal are expected to heavily strengthen their defence in the summer after their failings at the back were once again exposed in yesterday’s 3-0 Carabao Cup final defeat to Manchester City.
The Gunners have conceded the most goals out of the Premier League’s top six.
Per Mertesacker is retiring at the end of the campaign, Shkodran Mustafi has failed to impress while there are question marks over Calum Chambers and Rob Holding.
Gimenez has two-and-a-half years left on his contract with Atletico Madrid.
He has made 26 appearances in all competitions for Los Colchoneros this term.