The Daily Mirror carries reports from German newspaper Bild that suggests Liverpool boss Brendan Rodgers wants to act quickly to win the race for the talented Barca academy graduate.
And Rodgers is believed to be leading the chase for the Spain under-21 international, with Bayern Munich and Bayer Leverkusen also in hot pursuit.
Liverpool have switched to a 3-5-2 system in recent weeks, with new signing Mamadou Sakho in central defence alongside first-choice duo Martin Skrtel and Daniel Agger.
With Rodgers favouring this system, it's unlikely either Andre Wisdom or Martin Kelly would be trusted in the right-wing-back role.
Both Liverpool academy players played centre-back before switching to right-back upon graduation to the first-team. But neither are forward thinking full-backs - something Montoya most certainly is.
Obviously, Glen Johnson remains Liverpool's undisputed first-choice right-back, but the 22-year-old Montoya may be viewed as a potential long-term successor and useful back-up for the time being.
He's a product of the famous La Masia Barcelona youth academy, and has made 25 league appearances for the Catalan giants since his first-team La Liga debut in 2011.