According to Sport, the club planned to send a representative to Buenos Aires for Friday's meeting between Argentina and Uruguay at La Bombonera, with both defenders on the scouting list.
The search reflects the determination of sporting director Deco and Flick to add another central defender during the winter window. Barcelona have been assessing several candidates in recent weeks, including Stuttgart's Finn Jeltsch, Twente's Ruud Nijstad and Tottenham's Luka Vušković.
The club have also explored the possibility of bringing Bayern Munich defender Jonathan Tah to Barcelona on loan, although that approach has not produced an agreement.
Martínez offers the experience and immediate defensive reliability Barcelona are looking for. The 28-year-old Argentina international has been linked with a possible departure from Manchester United, where his contract is reportedly due to expire at the end of the season.
At 1.75 metres, he is not a physically imposing centre-back by conventional standards, but he compensates with aggression, anticipation and his ability to compete in individual duels. His experience at the highest level would also give Flick an established option rather than another young player requiring time to develop.
The arrival of Mauricio Pochettino as United head coach could, however, change the picture. The Argentine's appointment creates uncertainty over whether Martínez's situation at Old Trafford will improve, potentially making a move more difficult to negotiate.
Rodríguez represents a different type of investment. The 21-year-old left-footed defender has established himself at Cagliari since moving from Uruguayan club Peñarol and is regarded as one of the more promising young centre-backs to emerge from Uruguay.
He has become a regular starter in Italy and has also progressed into the senior national-team picture, although he remains behind more established options such as Ronald Araújo and José María Giménez.
Sport reports that Rodríguez could be available for around €15 million. Cagliari are understood to have paid approximately €4.5 million to sign him from Peñarol just over a year ago, meaning a sale would represent a significant increase on their original investment.
The two candidates therefore offer Barcelona different solutions: Martínez could strengthen the defence immediately, while Rodríguez would represent a longer-term project at a comparatively affordable price.
The January operation remains difficult because of Barcelona's financial restrictions, but the club's scouting activity in South America suggests the search is continuing beyond the European market.