The 26-year-old played an important role in Spain winning Euro 2024, scoring three goals and registering two assists in six appearances at the competition.
Olmo did not start the tournament in Spain's first XI, but he benefitted from an injury to Pedri in the quarter-finals of the competition to play a key role in the latter stages.
There has been widespread speculation surrounding the forward's future this summer, with a number of clubs believed to be keen to secure his signature, including Premier League giants Chelsea.
However, Barcelona are thought to lead the race for the attacker, who represented the Catalan outfit's youth academy between 2007 and 2014 before leaving for Dinamo Zagreb.
Barcelona 'place' six-year contact offer on table
According to Mundo Deportivo, Barcelona have offered Olmo a six-year deal to return to Camp Nou, and the Catalan outfit are now confident of winning the race for his signature.
The report claims that Leipzig have also agreed to keep the €60m (£50.6m) release clause in his current deal 'for a few more weeks'.
The buy-out fee is thought to have officially expired last week, but Barcelona are allegedly expecting to close a deal for €60m (£50.6m) following positive discussions with his German club.
The Catalan outfit would allegedly pay the amount over a longer period in order to ensure that they comply with financial rules.
Olmo is reportedly yet to make a final call on his future, as Liverpool, Atletico Madrid and Manchester United could yet place rival bids on the table.
Do Barcelona still want Williams?
Barcelona are also working on a deal for Athletic Bilbao attacker Nico Williams, who starred alongside Olmo for Spain at this summer's European Championship.
The 22-year-old scored twice and registered one assist in six appearances for his country at the competition, including a goal in the final against England.
There is also said to be Premier League interest in the forward, but he has allegedly given priority to Barcelona.
Williams entered the European Championship off the back of scoring eight goals and registering 19 assists in 37 appearances for Athletic during the 2023-24 campaign.
The attacker has a €58m (£49m) buy-out clause in his contract at San Mames.