The 29-year-old is expected to arrive in Porto Alegre on Sunday morning to finalize the switch, with the Gunners able to cut the move short after six months if they desire.
"[Santos] is coming from Arsenal to Gremio. Naturally everyone is talking a lot about [Hernan] Barcos, but Andre is a huge signing for us," Rui Costa said.
"They are both players who want to be part of our project, who come to add a lot of depth to our squad and to win titles."
With Kieran Gibbs sidelined for up to six weeks, boss Arsene Wenger acted quickly in the final days of the transfer window to recruit Malaga left back Nacho Monreal.
And Santos has now been allowed to find more regular first-team opportunities back in his homeland.