The assault shook German football ahead of crucial Champion League ties this week.
The assault, described by Dortmund city's police chief as a "targeted attack" against the team, shook German football ahead of crucial Champion League ties this week.
Investigations will focus on a letter claiming responsibility for the attack that was found close to the site of the blasts.
"The letter claims responsibility for what happened," prosecutor Sandra Luecke said late Tuesday, telling journalists that "its authenticity is being verified".
German authorities have held off from describing it as a terror attack, saying that it is too early to determine the motive.
But Germany has been on high alert since a series of jihadist attacks last year, including the Christmas market truck assault in Berlin in December that claimed 12 lives.
The explosives detonated minutes after Borussia Dortmund's team bus pulled away Tuesday from the squad's hotel and headed for their quarter-final, first-leg, tie against Monaco.
Spanish international Marc Bartra underwent surgery on a broken wrist after he was hit by flying glass.
The quarter-final match will now be played on Wednesday evening, just hours before another Champions League clash in Germany between Bayern Munich and Real Madrid.
"We are assuming that they were a targeted attack against the Dortmund team," said the western German city's police chief Gregor Lange, adding however that it did not amount to an organised terror assault.
The bus had set off for the Borussia stadium about 10 kilometres (six miles) away when "three explosive charges detonated" police said.
The explosives, which went off shortly after 7:00 pm (1700 GMT), were hidden in a hedge and were set off as the bus passed.
The blast shattered the bus windows and the vehicle was burned on the right hand side.
"The bus turned onto the main road, when there was a huge noise -- a big explosion," Dortmund's Swiss goalkeeper Roman Burki told Swiss media.
"After the bang, we all crouched down in the bus. Anyone who could, threw himself on the floor.
"We did not know if more would come."
Burki said Bartra was "hit by splinters of broken glass". Dortmund's press spokesman said the 26-year-old had broken the radius bone in his right wrist.
The club said other players were safe and there was no danger inside the Signal Iduna Park stadium.
"The whole team is in a state of shock, you can't get pictures like that out of your head," Dortmund CEO Hans-Joachim Watzke said.
"I hope the team will be in a position to be able to compete tomorrow on the pitch.
"In a crisis situation like this, Borussia pulls together."
Germany's best-selling Bild daily quoted anonymous sources saying that investigators were hunting for a likely getaway car used by the attacker.
The vehicle had foreign car plates, said the newspaper, which also added that police believed the explosives were a particular type of pipe-bomb.
The announcement that the game was postponed was only made to the stunned stadium about 15 minutes before Tuesday's match was due to start.
Dortmund's president Reinhard Rauball said he believed the team would be ready for Wednesday's game.
"The players will be able to push this out of their minds and be in a position to put in their usual performances," he said.
"The worst thing would be if whoever committed this attack was now able to get to affect them through it."
But ex-Dortmund player Steffen Freund, who won the Champions League with Borussia in 1997, said there would be scars.
"When there has been a direct attack on the team bus, then it's not just forgotten by Wednesday," said the 47-year-old.
"Mentally and psychologically that is hard to absorb, it's a lot to deal with."
Dortmund police said security would be tightened at Wednesday's match, with a major deployment of officers to "ensure that the game is played safely".
Separately, security was also being tightened at the Bayern-Real tie in Munich.
Bild said both teams' hotels were under heavy police guard, and the squads' buses driven to a safe location.
Germany has been on a high alert since last December's attack in Berlin, when a Tunisian national hijacked a truck and rammed it into a crowd, killing 12 people.
The German national team, which included some Dortmund players, was also in the Stade de France in Paris when jihadists attacked the French capital in November 2015, leaving 130 dead.