Antonio Conte's side won the Premier League title and could claim the double by winning Saturday's FA Cup at Wembley. Thousands of fans were expected to turn out for the parade.
"In light of these tragic events, we feel it is inappropriate to go ahead with the victory parade in London on Sunday," the club said in a statement.
"Given the heightened security threat announced by the (British) government, and recognising that this is a developing situation, we have given this careful consideration," it said.
"We strongly believe, in the interests of everyone, this is the correct course of action. We are sure our fans will understand this decision."
The club said black armbands would be worn by the players in Saturday's final against Arsenal and that a donation would be made to a fund supporting the victims.
Arsenal later announced that they were cancelling a planned screening of the final, to have been staged at their Emirates Stadium. The club also said there would be no victory parade were they to beat Chelsea.
"As always, the safety of our supporters and staff is paramount and we are in close contact with security services," chief executive Ivan Gazidis said.
"After taking their advice we have reluctantly taken the decision to cancel the screening and potential parade. We are sorry for any disappointment this causes but it is in everyone's best interests."
An Islamist suicide bomber blew himself up and killed 22 people at a packed pop concert at Manchester Arena on Monday.
On Tuesday Prime Minister Theresa May's government announced that the security threat level had been raised to "critical" as the bomber, Salman Abedi, had probably not acted alone.
(Reporting by Martyn Herman; editing by Mark Heinrich)
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