The Catalan outfit spent a chunk of the recent market attempting to sign a new right-back, and they eventually secured a loan deal for Manchester City defender Joao Cancelo in the closing stages of the window.
Cancelo was thought to have always been Barcelona's number one target in that area of the field, but according to Marca, Xavi's side had initially attempted to secure Molina's signature.
The report claims that the La Liga champions offered multiple players as part of a player-plus-cash deal, but Atletico turned down each offer that arrived for the Argentina international.
Barcelona were therefore forced to switch their attention elsewhere due to their financial problems, which prevented them from spending big money during the recent window.
Molina started his professional career with Boca Juniors before moving to Udinese, featuring on 68 occasions for the Italian outfit ahead of a switch to Atletico.
The 25-year-old impressed for Diego Simeone's side during the 2022-23 campaign, scoring four goals and registering four assists in 43 appearances in all competitions.
Molina missed Atletico's first two matches of this season through injury but was back for the last game against Rayo Vallecano, finding the back of the net in his team's 7-0 success.
The defender is currently with the Argentina squad, meanwhile, and he played the full 90 minutes of his country's World Cup qualification match against Ecuador on September 8.
Barcelona's financial problems made it a difficult window for the club, but five new players arrived, with Joao Felix being confirmed as a loan signing from Atletico on deadline day alongside Cancelo.
Oriol Romeu, Ilkay Gundogan and Inigo Martinez also moved to Camp Nou, but there were some notable departures in the shape of Ousmane Dembele, Sergio Busquets and Jordi Alba.
Ansu Fati was also allowed to make the move on loan to Brighton & Hove Albion, with the Spaniard seeing his career stall due to some injury problems.
Barcelona have made a strong start to their 2023-24 La Liga campaign, picking up 10 points from four matches, which has left them in third position in the table.
The Catalan giants will be bidding to make it four straight league wins when they continue their campaign at home to Real Betis on September 16.