The 35-year-old centre-back will return to Europe after a successful spell with Leon in which he won the Liga MX Apertura in 2013 and the Clausura this year.
Marquez, who appeared at his fourth World Cup in Brazil, won four league titles during his time a Barcelona before he was released four years ago.
A two-year stay in the United States with New York Red Bulls followed before he returned to his homelande with Leon.
The vastly experienced defender, who has 124 Mexico caps, is now set to write another chapter in Italy with Verona.
Leon revealed on their Twitter account on Monday: "Rafael Marquez has been transferred to Hellas Verona in Serie A Italiana. We hope that this new stage is full of successes."
News of Marquez's arrival comes on the same day that Verona brought in Aston Villa midfielder Antonio Luna on a season-long loan deal.
Luna moved to Villa Park at the start of last season from Mallorca and made 17 appearances in the Premier League, but is not part of Paul Lambert's plans.
