The current vice president has the backing of 2,214 delegates â well beyond the simple majority of 1,976 needed according to a survey by Associated Press.
But the poll is unofficial and the Democrats' official candidate will not be known until all delegates who are the elected representatives in their area â have voted at next month's party convention.
Ex-district attorney Ms Harris hit the campaign trail in Milwaukee last night her second public appearance since Joe Biden dropped out of the presidential race.
The Wisconsin city is where Mr Trump became the Republicans' official candidate last week.
In a blistering attack on her rival, Ms Harris told cheering supporters that as a prosecutor she took on "predators who abused women, fraudsters who ripped off consumers, cheaters who broke the rules for their own game".
"So hear me when I say, I know Donald Trump's type. In this campaign I will proudly put my record against his any day of the week," she declared. The audience of around 3,000 supporters erupted in cheers and chants of "Kam-a-la".
Ms Harris accused Mr Trump of aiming to cut Social Security and Medicare, while planning tax breaks for billionaires and promising oil lobbyists "he would do their bidding" for $1billion in campaign donations.
She insisted: "We are not going back", which the crowd instantly picked up as a chant.
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