The 35-year-old, who has just graced his fourth World Cup finals, is currently back in his homeland with Club Leon.
Prior to heading home last summer, Marquez had spent the previous two years in the MLS with the New York Red Bulls.
Four years have now passed since he last played in Europe, as a successful seven-year stint at Barcelona came to a close, but a fresh start in Italy may appeal to Marquez.
Hellas Verona acknowledge that it will be difficult to secure the services of such a high-profile figure, but they will explore the possibility before turning their attention elsewhere.
Director of football Sean Sogliano told Sky Italia: “Marquez? I have a lot of dreams, and then I wake up.
“We will see. The window is long.”
Speculation is also suggesting that Hellas Verona are looking to take Aston Villa defender Antonio Luna to Serie A.
The Spanish full-back struggled to make an impact during the 2013/14 campaign, his first in England, and has been omitted from Villa’s squad for a summer tour to Texas.
Talks with Hellas Verona are reported to have already been opened, with Paul Lambert ready to move on those who no longer figure in his plans