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Prime Minister Sir Keir Starmer said at a news conference on Monday: "Liverpool stands together and the whole country stands with them".
Former Liverpool manager Jurgen Klopp, who attended the parade, said the incident marred what "should have been one of the greatest days in the history of the city".
Speaking on Tuesday after he was inducted into the League Managers Association Hall of Fame 1,000 Club, the German added: "It just showed the two faces of life.
"The most beautiful face for a long time: the parade was incredible, the mood was incredible.
"And from one second to another everything changed and we learned again there are more serious things in the world than football.
"Thoughts and prayers go to the injured people and their families as well."
Ex-Liverpool manager and player Kenny Dalglish said he was "shocked, horrified and deeply saddened" by the incident.
Liverpool's chief executive officer Billy Hogan said: "I would like to extend our heartfelt thoughts to all those who've been affected by this appalling incident.
"This weekend was one of celebration, emotion and joy spread across the city in our entire fanbase, and it ended in unimaginable scenes of distress with this appalling incident."
Tens of thousands of supporters were on the streets of Liverpool on bank holiday Monday to celebrate the side winning the Premier League.
Water Street was closed to traffic for the parade.
However, eye witnesses said the car ignored 'road closed' signs.
The car ploughed into supporters at about 18:00 BST, approximately 10 minutes after the bus containing Liverpool's players, staff and the Premier League trophy had departed.
Police say they are not treating the incident as terror-related.
Eleven people remained in hospital as of Tuesday afternoon.