The 22-year-old, who was a regular under former manager Tito Vilanova last season, has played in just two league matches this season under Tata Martino.
Montoya's contract runs out at the end this season and he's been getting no first-team football under new coach Gerardo Martino.
The Liverpool board have a policy to recruit the best young talent and 22-year-old Montoya would fit into their future plans.
The Barcelona star is yet to make a decision on his future but finds the prospect of joining Liverpool appealing, according to sources in Spain.
For the European Under-21 champion, it must be immensely frustrating to find opportunities so hard to come by, having made good initial progress in his 15 league appearances last campaign.
However, Montoya finds himself competing with the excellent Dani Alves for a place in the Catalan club's starting XI, and the Brazilian has begun the current campaign in excellent form.
Barcelona, though, would be making a mistake in allowing a home-grown Spanish youth international of some repute head for the exit door. If they do not hand him adequate first-team opportunities, that is exactly the scenario that is developing.
Montoya will not sign a new contract if he is not going to play regularly. He could be a first-team regular at several historic European clubs, and there is plenty of interest in his contract situation—not least from the Premier League.
It is a difficult situation for Martino, who clearly must choose his better player, Alves, for big matches. However, it is debatable how much longer the Brazilian will remain in his prime or, indeed, at Barcelona having turned 30 earlier this year.
Montoya's situation is not a new conundrum for the club, who lost the player's international colleague Thiago Alcantara to FC Bayern only this summer after the midfielder became unhappy with his lack of playing time.
Barcelona have world-class players in almost every position. It takes an extremely bold manager to show faith in a youngster like Montoya, at the expense of arguably the world's best right-back in Alves.
Martino must quickly weigh whether he sees Montoya in his first-team plans for next season. Should he envisage Alves staying on at the club, the youngster will doubtlessly choose to move on.
In Brendan Rodgers' Liverpool, he could find a perfect match in a young, talented side playing a familiar brand of possession-based football. Under Bosman rules, he would be free to sign a pre-contract in January, per Nick Lustig of the Daily Star.
Liverpool currently have Glen Johnson as a right-back option, as well as Martin Kelly, but the latter is seemingly yet to win over the Anfield boss after lengthy injury problems. The technically gifted Montoya, who is more comfortable on the ball, is likely more in line with Rodgers' vision.