The 19-year-old, born in 2006, spent last season on loan at Swiss club Grasshopper, where he scored nine goals and provided six assists in 36 appearances across all competitions, performances that have brought him to the attention of Lazio's recruitment department.
According to Gazzetta dello Sport, negotiations between the two clubs are close to being finalised, with a deal potentially able to be agreed within the coming hours. However, any official confirmation may not arrive until July, following a similar pattern to Lazio's earlier move for defender Pedraza.
Lazio are reported to have an agreement in place with both Asp Jensen's representatives, who also act for Lazio players Gustav Isaksen and Oliver Provstgaard, and with Bayern Munich over the structure of the deal.
The fee being discussed is in the region of €4m, with Bayern Munich retaining a sell-on percentage that could range between 30% and 40% of any future transfer.
For Gattuso, who is building his squad for his first season in charge at Lazio, the addition of a young, attacking midfielder with a productive loan season behind him represents the kind of forward-looking signing that fits a club working with financial constraints but seeking to add quality to its attacking options.
Asp Jensen's path mirrors a wider trend in recent transfer windows, with Serie A clubs increasingly looking to sign promising young players from major European academies who have proven themselves on loan at a lower level before stepping up.
Should the move be completed as expected, Asp Jensen would arrive at Lazio with significant room for development, joining a squad being reshaped under Gattuso's direction. The structure of the deal, including Bayern Munich's retained interest in any future sale, reflects the value the German club continues to see in a player they are nonetheless willing to let leave at this stage of his career, betting that Lazio represents the right environment for his next step.