"Coudet is confirmed (as the new coach)," River president Stefano Di Carlo told local television, adding that the Argentine manager would be officially presented on Wednesday.
Coudet joins River, where he played in the early 2000s, from Spain's Alaves.
The 51-year-old coach is no stranger to the Argentine top flight, having led Rosario Central and Racing Club, as well as Mexican side Tijuana, LaLiga's Celta Vigo, and Brazil's Internacional and Atletico Mineiro.
Coudet will replace Gallardo, who resigned last week after a second spell at the club lasting a year and a half, during which he failed to win any titles.