Arthur, 21, will join fellow Brazilians Paulinho and Coutinho in the Barcelona midfield when the deal is completed this summer.
“The club’s doctors told me I’ll be back with the team in approximately two weeks. Now it’s time for a full recovery!”
In Aguero’s absence, City’s attack is likely to be led by Gabriel Jesus. Their next game is at Everton on March 31.
Argentina will play Italy at the Etihad Stadium in Manchester on March 23 and then face Spain in Madrid four days later as part of their World Cup preparations.