Recent reports claim that the Catalan giants are making progress in their pursuit of the 30-year-old, who will see his contract at Betis expire at the end of the season.
After three years in Mexico with Club America, Rodriguez moved to Europe to join Betis in 2020 and has since established himself as a key player for the Spanish outfit.
The deep-lying midfielder has made a total of 170 appearances for Betis across all competitions, including 21 La Liga outings so far this season.
Rodriguez most recently played in a 1-1 draw with city rivals Sevilla on Sunday, leaving Manuel Pellegrini's side seventh in the table and just two points behind the top six with five games remaining.
Rodriguez "very happy" at Betis amid Barca links
Rodriguez, a 29-cap Argentina international who won the 2022 World Cup, has allegedly been identified by Barcelona as a potential replacement for Oriol Romeu, who has struggled for regular game time under head coach Xavi.
Despite those links with Barca, Rodriguez has insisted that he is "very happy" at Betis and neither he or the club have made a decision over his future.
"I feel at home at Betis. I feel very comfortable here, I'm very happy." Rodriguez told reporters as quoted by 90min. "I've been here for four and a half years and the truth is that I'm happy, very happy and enjoying these last remaining games of the season.
"You have to make decisions in life and well, we'll see what happens at the end of the season. The truth is that as of today, neither I nor the club have any certainty about what might happen.
"There is not a single option on which my decision depends. It depends on many things. I've already said it, it's a recurring theme and I don't want to create any false expectations or say anything for people."
Barcelona exploring free agent market once again
Amid their well-documented financial troubles, Barcelona have been no strangers to recruiting players on free transfers and they are expected to continue with this approach ahead of the 2024-25 season.
Last summer, the Catalan giants successfully lured midfielder Ilkay Gundogan from treble winners Manchester City on a free transfer, as well as defender Inigo Martinez from Athletic Bilbao.
In 2022-23, Barca snapped up both Andreas Christensen and Marcos Alonso from Chelsea, while Franck Kessie and Hector Bellerin - who have since left the club - joined from AC Milan and Arsenal respectively.
Barcelona are also expected to delve into the loan market as they did last summer, and will hope to negotiate deals to keep hold of both Joao Cancelo and Joao Felix, either by extending their current loans from Manchester City at Atletico Madrid respectively or by signing them on permanent deals.