The pair were among four Premier League players accused of not following Covid protocols upon entering Brazil.
Villa had previously agreed with Argentina that both Martinez and Buendia would be available for selection for games against Venezuela and Brazil.
They were then due to miss Argentina’s third game against Bolivia.
It is understood Martinez and Buendia have now flown to Croatia rather than the UK. Flying straight to the UK from South America would have meant 10 days in quarantine.
But stopping for that period in a green list country means they can train and then travel back to the UK without quarantining.
Both men will now miss Villa’s game with Chelsea on Saturday but they should be clear to face Everton on September 18.