Broadcaster Sky, top-selling daily Bild and magazine Kicker say Tedesco, 36, is on the verge of agreeing to take on the job.
Marsch was sacked last Sunday, after just four months in charge, due to a poor series of results.
RB Leipzig have finished in the top three in the past three seasons but presently sit 11th in the table and trail leaders Bayern Munich by 16 points.
Tedesco was born in Italy, but grew up in Germany.
He steered Schalke to second in the Bundesliga at the end of the 2017/18 season, but was sacked in 2019 following a downturn in their form.
He has spent the last two seasons coaching Spartak Moscow.
On Tuesday under caretaker coach Achim Beierlorzer, RB Leipzig pulled off a shock 2-1 home win over Manchester City in the final round of Champions League group-stage matches.
Despite the result, the English giants still progressed to the last 16 as Group A winners while the Germans, who finish third in their group, drop into the Europa League.