The 29-year-old arrived at Arsenal in 2011 in a £6.8million move from Fenerbahce but has fallen down the pecking order at the Emirates.
Kieran Gibbs has been Arsene Wenger's first-choice at left-back this term and the arrival of Spaniard Nacho Monreal in the transfer window has further reduced Santos' chances of selection.
There has been no official comment from Arsenal regarding Santos' exit, but he is believed to be on the brink of signing for Brazilian side Gremio.
Arsenal striker Lukas Podolski earlier posted on Twitter: "@Andre--santos27 We will miss you man, I wish u all the very best #Gunner God bless you."
It is unclear whether the transfer will be permanent or a loan move.