This is according to reports from Spain cited by The Daily Express, which report that the Reds recently had an £18 million bid for Southampton's Lovren rejected, with the club supposedly now desperately keen to do business with the Catalan giants to send Agger away from Anfield.
Indeed, the same source further claims that Barcelona - who are now managed by former Celta Vigo boss Luis Enrique - first received a potential proposal concerning Agger several months ago during the final throws of the Gerardo Martino era as they began the search for an adequate replacement for retiring club stalwart Carles Puyol.
This particular proposal was rejected at that time, however, and it is not clear if Barcelona are more willing to pursue a deal for £12 million-rated Agger now they are under new management.
Agger to join Suarez at the Camp Nou?
While it is claimed that such a proposition was not in connection with any potential deal for Luis Suarez, one can't help but think that may become a distinct possibility if a straight-cash deal for the controversial Uruguayan international cannot be fully agreed soon.
Suarez - who looks set to depart the Premier League following his most recent shameful biting episode involving Italy defender Giorgio Chiellini - is believed to be the current subject of advanced talks between the two European heavyweights, although it is not yet apparent if a suitable and concrete arrangement has been decided upon.
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A departure from Liverpool would presumably be a very sad event for Agger, given that the 29-year-old has now spent eight years of his career on Merseyside.
First brought to Anfield by Rafael Benitez in 2006, the former Brondby starlet became a mainstay of the defence and helped the team to lift the FA Cup in 2006, the Community Shield in the same year and indeed the League Cup in 2012.
Agger was also part of the Liverpool side who finished runners-up in the 2007 Champions League following a 2-1 final defeat to AC Milan in Athens.
Last season saw the end of Agger's consistent run of starts, however, with the experienced Dane seeing his influence gradually reduced following the respective arrivals of Kolo Toure and Mamadou Sakho.
He made just 20 appearances in total during a 2013/14 season in which Liverpool were renowned for their phenomenal attacking prowess but ultimately were denied their first top-flight title of the Premier League era due to a number of desperately poor defensive displays.
Is he good enough for Barcelona?
Given that he is seemingly no longer indispensable at Liverpool, a move to Barcelona would appear to be a very positive outcome indeed for Agger.
Whether he is still good enough to play for such a reputable club with obvious ambitions of domestic and European dominance, however, remains to be seen.
With the charismatic Puyol now having retired and former Liverpool midfielder Javier Mascherano not having always appeared entirely comfortable alongside Gerard Pique in his makeshift defensive role, there can certainly be no question that Enrique will need to improve his options at the back as a matter of urgency.