The incident on 28 July is understood to have happened because of low water levels on the river.
The 190-passenger Ullur, according to Viking, came into contact with a sandbank. There were 186 passengers and 53 crew on Ullur at the time.
A Viking spokesperson said: "The safety and wellbeing of our guests and crew is always our highest priority.
"There were no injuries to guests or crew, and the vessel remained safe throughout the incident.
"Guests have been safely transferred ashore and will continue their planned itinerary with a full day in Bucharest before concluding on Thursday as scheduled."
Cruise writer Sara Macefield recently sailed on the Danube with Uniworld but the Vineyards and Palaces of the Danube cruise was impacted by low water levels.
Writing for TTG, she revealed the worst-affected stretch was through Eastern Europe, where the waterway has fallen to levels not seen since 1996.