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.
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."
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- Arsene Wenger
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