The Athletic was the first to report Miami's interest.
Martinez, 25, would be Miami's second Designated Player signing following the acquisition of 19-year-old Estudiantes winger Matias Pellegrini.
The Colombian has scored 17 goals in 65 league and cup appearances for Las Aguilas since he was acquired from La Liga side Villarreal in 2018. Prior to his stint with Villarreal, Martinez was on the books of Chinese Super League side Jiangsu Suning where he scored 18 goals in 36 league and cup appearances. He made his professional debut with Racing Club in 2013, before being loaned out to Argentine clubs Deportivo Santamarina and Aldosivi.
At international level, Martinez has made 18 appearances for Colombia, scoring two goals.
A source confirmed that Miami had also been pursuing River Plate attacker Exequiel Palacios, but reports out of South America indicate that Bundesliga side Bayer Leverkusen has the inside track to obtain Palacio's signature.
A source also told ESPN that Miami are in the market to acquire Sebastian Driussi from Zenit St. Petersburg, with the odds of a deal being "50-50."