Dortmund confirmed Rose had agreed to be its new coach.
“Out of respect for all the parties involved, after today Borussia Dortmund will only comment again on the sporting future under Marco Rose’s leadership after the end of the 2020-21 season,” the club added.
Dortmund fired Lucien Favre in December and put Edin Terzic in interim charge.
The team has won only four of 10 league matches under Terzic — and one of its last six — ahead of Wednesday’s Champions League last-16 game at Sevilla.
Rose won plaudits when Gladbach led the Bundesliga for much of the first half of last season before eventually finishing fourth and qualifying for the Champions League.
Gladbach hosts Manchester City in the Champions League next week in a game moved to Hungary because of German travel restrictions during the coronavirus pandemic.
Dortmund is sixth in the Bundesliga and Gladbach is seventh. The two clubs are also due to meet in the German Cup on March 2.