Koeman, as it stands, is still in his position at Camp Nou, but there could be a change at the helm during the upcoming international break.
Legendary midfielder Xavi and Belgium boss Roberto Martinez have both been linked with the job at Barcelona, but according to RAC1, the La Liga giants have already touched base with Gallardo.
The report claims that 45-year-old is not prepared to terminate his deal early with River, but his contract is due to expire at the end of December, so he would be open to a move to Spain in January.
Gallardo, who turned out for the likes of Monaco and Paris Saint-Germain during his playing days, has been in charge of Los Millonarios since 2014.
Barcelona have allegedly decided to keep Koeman in his role for Saturday's La Liga clash with Atletico Madrid, although the Dutchman will be in the stands, as he serves the second of a two-match touchline suspension.