Dortmund sacked Lucien Favre as manager in December and have since been coached by Edin Terzic on an interim basis.
"Over the past few weeks, we held several talks on Marco's future," Gladbach sporting director Max Eberl said in a statement. "Sadly, he has now decided to trigger a clause in his contract valid until June 2022.
"He wants to make the move to Borussia Dortmund in the summer."
Sources told ESPN that Rose informed his squad of his decision on Sunday before their 0-0 draw at Wolfsburg.
Dortmund CEO Hans-Joachim Watzke confirmed to German wire SID that Rose will take over in the summer but refused to make any more comments.
"We are delighted," he said. "But there is not more to say at the moment. We will do that when he takes over in the summer. Also out of respect for all involved."
Both clubs are competing for a Champions League place in the Bundesliga and will meet in the DFB-Pokal quarterfinals on March 2.
Gladbach are seventh in the Bundesliga standings -- only one place behind Dortmund, who are sixth courtesy of their superior goal difference. Both sides are six points behind fourth-place Wolfsburg in the final Champions League qualification spot.
"Until [the end of the season], we will work with Marco, mobilise all our forces to achieve our goals in Bundesliga, the DFB-Pokal and the Champions League," Eberl said.
It has been a difficult season for Dortmund who were tipped as title contenders at the beginning of the campaign but now face a battle to finish in the top four. Terzic has so far picked up 14 points from his first 10 Bundesliga games in charge.