The 23-year-old Chile international, who had previously been linked with interest from Arsenal and Sao Paulo, joins Gremio for a loan fee in the region of £1.6m.
"Eduardo Vargas will wear the blue, white and black of Gremio," read a statement on the Brazilian club's official website.
"Vargas will leave Italy and will arrive at Porto Alegre tomorrow.
"After being welcomed at the airport by the supporters, he will officially be unveiled at the Olympic Stadium."
Vargas has been capped 15 times at senior international level, scoring two goals.