The Portugal international has again been a regular for Pep Guardiola's side this season, making 52 appearances in all competitions, including 34 outings in the Premier League.
The 25-year-old has struggled to show his best form during stages of the 2019-20 campaign, though, and he was left out of the starting XI against Real Madrid in the Champions League on Friday night.
According to The Telegraph, Barca are keeping a close eye on the developments surrounding the versatile attacker and could make an attempt to sign him during this summer's transfer window.
The report claims that the Catalan giants would struggle to afford the Portuguese in a straight cash deal but could offer players as makeweights, with right-back Nelson Semedo a potential option.
Bernardo moved to the Etihad Stadium from Monaco in the summer of 2017.