Reds boss Brendan Rodgers has been on the look-out to strengthen his back line and the current injury problems being suffered by Glen Johnson highlights the need for extra cover, especially in the right-back area.
The 22-year-old Montoya has seen his route to the first team at Camp Nou blocked by Daniel Alves and he is now said to be considering his future and he enters the final year of his contract.
His agent, Jose Maria Orobitg, says his client has yet to be offered a new deal and could be open to a move away either in January or next summer.
"For the moment he is under contract with FC Barcelona until the end of the season," he revealed. "However, the contract extension has still not arrived.
"We have talked but there is no certainty that Martin Montoya would hold down a first-team spot, because right now Daniel Alves closes the door.
"The transfer window is closed, but I can tell you that the contract extension may not be successful. Anything could happen between now and June."