Santos has garnered plenty of criticism this season for some poor performances, with the Gunners signing Nacho Monreal from Malaga in the January transfer window.
The 29-year-old has made 36 appearances for Arsene Wenger's side since joining in the summer of 2011, scoring four goals.
Kieran Gibbs and Thomas Vermaelen are currently facing spells on the sidelines, while Monreal is unavailable in Europe, meaning that Bacary Sagna may be used as a makeshift left-back for the club's Champions League tie against Bayern Munich.
Gremio will be the fifth Brazilian club that Santos has represented after Corinthians, Atletico Mineiro, Flamengo and Figueirense, while his stay in Europe has taken him to Fenerbahce before joining the Gunners.
Santos' last appearance for Arsenal came in the 2-2 draw against Liverpool, while he was an unused substitute in the 1-0 win over Stoke.
Santos said he was excited ahead of a move to Brazil, but that he would need time to regain his match fitness.
"I have the best expectations," Santos told Lancenet. "[Gremio] are one of the biggest clubs in the country.
"I've come to Gremio to be a champion. I am very excited about this project. I have dreamed of working with Vanderlei Luxemburgo. He is a top manager in our football.
"If there was a game today, I would like to play.
"But I need to be match fit as I played a game one day, then not the next one. I'll need five or six games to be fitter. I'm anxious to know the squad."