UEFA said the match will now be played on Wednesday at 18.45 (local time).
Three explosions rocked the Borussia Dortmund bus, injuring Spanish international Marc Bartra, as the German team headed for a Champions League game against Monaco.
Bartra faces wrist surgery after being hit by flying glass and the quarter-final match was postponed until today because of the attack. The club said the other Dortmund players were "shocked" but safe.
"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.
Local prosecutors also revealed that a letter has been found close to the site of the blasts, which struck shortly after the team bus drove off from the squad's hotel.
"Its authenticity is being verified," prosecutor Sandra Luecke told journalists, without giving details on its contents.
The bus had set off for the Borussia stadium about 10 kilometres away when "three explosive charges detonated," said police.
The explosives, which went off shortly after 7.00pm (local time), 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 confirmed the 26-year-old "will be operated on, he has broken the radius (bone) in his right wrist and there are foreign bodies in the arm".
The club said other players were safe and there was no danger inside the Signal Iduna Park stadium.
Star striker Pierre-Emerick Aubameyang was seen being led away from the team bus through the traffic.
"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."
The announcement that the game was postponed was only made to the stunned stadium about 15 minutes before Tuesday's game was due to start.
UEFA said the match will now be played on Wednesday at 18.45 (local time), adding that the decision was made in consultation with the club and local authorities.
Dortmund relayed a copy of its Twitter announcement of the explosions inside the stadium to inform fans of events.
The club also urged fans in the stadium not to panic.
The stadium slowly emptied before Monaco players came on the pitch for a short training session.
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 today's match, with a major deployment of officers to "ensure that the game is played safely."
Separately, security was also being tightened at another major Champions League clash in Germany today -- between Bayern Munich and Real Madrid.
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.
Bartra's former team Barcelona quickly expressed its sympathies, tweeting: "All of our support to @MarcBartra, @BVB and their fans."
Spanish Prime Minister Mariano Rajoy also wished Bartra "a speedy recovery" on his Twitter account.