Top New York donors and Democratic fund-raisers, in more than two dozen interviews, said that their phones rarely stop buzzing as candidates blitz one of the densest concentrations of Democratic wealth in the country.

Others calling and visiting include Terry McAuliffe, the former Virginia governor; Mayor Eric Garcetti of Los Angeles; former Gov. Deval Patrick of Massachusetts; Gov. Steve Bullock of Montana; and the former Missouri Senate candidate, Jason Kander. Then there are the ambitious locals who already keep their donor lists close at hand: Gov. Andrew M. Cuomo and Senators Kirsten Gillibrand of New York, Cory Booker of New Jersey and Chris Murphy of Connecticut.

“When a candidate calls me to talk about ‘strategy and issues,’ you grab hold of your wallet for dear life,” said Robert Zimmerman, a prominent New York donor and a member of the Democratic National Committee who has been in touch with multiple prospective candidates.

For now, it is more about making connections than collecting cash, as few donors are committing at this stage. But to run a serious primary campaign, Democrats know they will have to amass tens of millions of dollars in the coming two years. Even if they got started as early as this July (and none of the top tier is expected to get into the race until after the midterms), a candidate would have to raise nearly $55,000 per day to construct a $30 million war chest by the end of 2019.