The BBC, Athletic, and Sky Sports have reported. United will initially pay €76.5 million for the 22-year-old, with an additional €8.5 million in performance-related add-ons.
Newcastle also expressed interest in signing the Slovenian, potentially to replace their star player who wishes to leave. Alexander Isak, a target for Liverpool.
However, Sesko has been attracted by the lasting prestige of the 20-time English champions, despite the absence of European football next season.
His signing will complete a significant transformation of the Red Devils' underwhelming attack.
Only four other teams scored fewer than United's 44 goals in 38 Premier League matches last season, as the English giants faced their worst campaign in 51 years, finishing 15th in the standings.
Matheus Cunha and Bryan Mbeumo were acquired earlier in the transfer window, bringing United's total expenditure on forwards in recent months to over £200 million.
Sesko had also been previously linked with Arsenal after making a strong impression during his two-year tenure in Germany, where he netted 39 goals in 87 appearances for Leipzig, including 21 last season.
At 1.95 meters (6 feet 4 inches) tall, Sesko's impressive stature, rapid speed, and goal-scoring ability have drawn comparisons to Manchester City's Erling Haaland.
Similar to Haaland, he also gained prominence at RB Salzburg before transitioning to Germany in 2023.
United is relying on Sesko to fulfill his vast potential and help them regain their position at the top of the Premier League table.
Together, Sesko, Cunha, and Mbeumo scored a combined total of 58 goals last season.
The substantial investment in their signings needs to yield quick results, as United faces pressure to comply with financial sustainability regulations, especially without the income from European football this season.