The American, who has also coached New York Red Bulls and Red Bull Salzburg, left the German club in December last year after a five-month spell.
The 48-year-old will succeed Marcelo Bielsa, who was sacked on Sunday, once a work permit is finalised.
Leeds sit 16th in the Premier League table, two points above the relegation zone, following a run of heavy defeats.
After leaving Major League Soccer side New York Red Bulls in 2018, Marsch first joined RB Leipzig as Ralf Rangnick’s assistant before being appointed at RB Salzburg in June 2019.
He enjoyed a successful spell in Salzburg winning a league and cup double in his first season.
He was also part of the development of striker Erling Braut Haaland before the Norway striker joined Borussia Dortmund in 2019.
He then returned to Leipzig in April last year, this time as head coach when he took over from Julian Nagelsmann who left to manage Bayern Munich.
Marsch, who has been linked with the managerial vacancy at Celtic in the past, completed his Uefa Pro Licence in Scotland.