Bartra was the only person hurt in the incident, with police detailing three explosions as the Dortmund bus left the hotel to travel to Signal Iduna Park.
Dortmund confirmed the centre-back underwent surgery on a fractured wrist and to remove debris from his arm.
The Bundesliga club tweeted: "Diagnosis of @MarcBartra: breakage and sprinklings of debris on the right wrist. He's currently in surgery. Good improvement, Marc!"
Police director Michael Stein noted: "The status of the player Bartra changed in course of the evening from slightly injured to seriously injured."
As a result of the attack on the Dortmund bus, the match was postponed, with UEFA announcing a planned rescheduling for 18:45 local time on Wednesday.
Speaking at a news conference, public prosecutor Sandra Lucke confirmed that the incident was being treated as attempted murder.
Lucke said: "The investigation is being carried out based on suspicion of attempted murder. A letter was found near the crime scene. We are checking its authenticity intensively.
"We found someone taking responsibility for the attack in the letter. Due to the investigation, we cannot comment further on this.
"Another object that did not explode was found. Our current point of view is that it was a dummy."
"The Royal Spanish Football Federation wants to show its total backing for the Spanish international footballer, Marc Bartra, after he was injured as a result of some explosions that affected the bus of his club, Borussia Dortmund," the governing body said in a statement.
"The RFEF wishes him a quick recovery and condemns any type of violence."
Goalkeeper Roman Burki was alongside Bartra when the attack happened and he told Blick of how Dortmund's players ducked for cover following the first blast.
"I was sitting in the back row next to Marc Bartra, who was hit by fragments of the broken window," Burki said.
"After the bang, we all ducked down on the bus and whoever could got on the floor. We didn't know what else was going to happen.
"The police were quickly on the spot and they handled the situation.
Dortmund's chief executive Hans-Joachim Watzke is confident the players will pull together ahead of the rescheduled tie with Monaco.
"The team and the coaches are of course in shock," Watzke told Sky. "We must now channel it in some way.
"It will not be easy to get it out of our mind. In such a crisis situation, all of Borussia Dortmund will pull together. I think the team will feel it tomorrow."
"We're all shocked. Nobody thought about a football match in those minutes."