The former Dinamo Zagreb starlet plied his trade in Barca's La Masia academy ranks from 2007 to 2014, but he left without managing to break into the first team at Camp Nou.
However, having become a key player for his club while rising to international fame for Spain - starring during their run to Euro 2024 supremacy this summer - Olmo is thought to be wanted back at Camp Nou.
The 26-year-old is understood to have agreed personal terms over a six-year contract until the summer of 2030, but Barcelona are yet to agree a fee with Leipzig, who have reportedly already turned down two Blaugrana offers.
The second bid could have hit a reported £55.2m through performance-related add-ons, but Barca were only offering £42.6m up front, whereas Leipzig are looking for a guaranteed £46.9m.
Rose tells Barcelona to meet Olmo "value"
Leipzig are also said to be unhappy with Barcelona's proposed payment structure, as Die Roten Bullen want to receive the full fee within a few months, whereas Barca want to postpone an instalment until next year.
The Bundesliga outfit hold all the cards at the negotiating table, as a €60m (£51.2m) release clause in Olmo's contract expired in the middle of July, and his terms run for another three seasons.
News of Barcelona's second rejected bid was recently put to Leipzig head coach Rose in a press conference, to which he responded that the Blaugrana must meet Olmo's "value" if they are to prise him away from the Red Bull Arena.
"The player has value. He earned it with us. He has developed brilliantly. Everyone knows what Dani Olmo can do. Accordingly, everyone must be clear about his value and under what conditions the club might be willing to talk," Rose said.
"I think it is also important for Dani that he is shown his worth and knows it. And the club that absolutely wants him should then also do justice to the value of Dani Olmo."
How has Olmo performed for Leipzig and Spain?
From 148 matches in the RB Leipzig strip, Olmo has conjured up a respectable 63 direct goal contributions across all tournaments, scoring 29 times and setting up another 34.
Injuries restricted the playmaker to just 25 matches in the 2023-24 campaign, but he had a huge impact when fully fit, netting eight goals and providing five assists, including a terrific hat-trick in a 3-0 DFL Super Cup win over Bayern Munich.
Ironically, it was an injury to potential future Barcelona teammate Pedri that provided Olmo with a route into the Spain XI at Euro 2024, where he was responsible for three goals and two assists, scoring against Georgia as a substitute before Pedri's early withdrawal in the quarter-final showdown with Germany.
The Leipzig man opened the scoring in that contest as a substitute and set up Mikel Merino's dramatic extra-time winner, before striking the crucial goal in Spain's 2-1 win over France in the semis.
Olmo was unable to score or assist for the fifth game running in the final against England, but he made a critical goal-line clearance from a late Marc Guehi header, celebrating that heroic moment as if he had scored himself.