It had already been noted by Sky Sport Italia and Sportitalia that the Aquile were targeting Brandt under new coach Maurizio Sarri.
Now Calciomercato.com claim the formal proposal has now been sent to the Bundesliga side.
It is a paid loan at a cost of €3-4m with option to buy at the end of the season for a total €25m.
However, that would become an obligation to make the transfer permanent if Lazio qualify for the Champions League.
The 25-year-old can play on the left of a trident attack or in an attacking midfield role.
Lazio face competition from Arsenal, who are also targeting Germany international Brandt.
He made 45 competitive appearances for Borussia Dortmund this season, scoring four goals and providing two assists.
They paid €25m to sign him from Bayer Leverkusen in 2019.
— calciomercato.com (@cmdotcom) June 28, 2021