Mystery still surrounds how the singer came to suffer the injury. Details regarding the accident have not been made available.
Bono is under the care of Dr Hans-Wilhelm Muller-Wohlfahrt, a world-renowned sports injury specialist who is the club doctor at Bayern Munich. He is also being treated by leading German neuro-surgeon Professor Dr Jeorg Tonn.
Tour rehearsals were to begin tomorrow for the opening date in Salt Lake City, Utah, on June 3.
That concert has since been postponed. Bono is expected to return to Ireland to recuperate.
A statement from the band said further information about other tour dates would be announced in due course, once the singer's condition had been assessed.
U2 manager Paul McGuinness said: “We will make plans to reinstate the dates as soon as possible.” He added it was “unfortunate that we're inconveniencing fans”.
U2 are the headline act at Glastonbury, which is celebrating its 40th birthday this year. However, that show on June 25 is also in doubt.
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