Santos only recently returned to Emirates Stadium after spending the second half of last season on loan at Gremio.
The 30-year-old is thought to be keen on returning to Brazil and Flamengo are trying to agree a deal with the Gunners for the left-back.
Flamengo coach Mano Menezes is an admirer of Santos having worked with him while he was in charge of Corinthians, but the Brazilians face a race against time to sign the full-back with their transfer window closing on July 20.
Flamengo's director of football Paulo Pelaipe admitted they are waiting to hear back from the Premier League side if they will sanction the deal.
"The negotiation is well under way, but we have the transfer window problem as it's about to close," Pelaipe told UOL. "We are waiting for a signal from Arsenal.
"I don't want to talk about optimism or pessimism, as I don't want to make the fans create expectations.
"But if Arsenal accepted to talk with us and listen to offers, I think we won't have problems, let's wait."