Former Vice President Joe Biden told at least a half-dozen supporters Tuesday he intends to run for president and asked for their help in lining up contributions from major donors so he can quickly raise several million dollars, a person familiar with the matter said.

Mr. Biden has expressed concern to these people that he wouldn’t be able to raise millions of dollars in online donations immediately the way some other Democratic candidates have, including former Rep. Beto O’Rourke of Texas and Sen. Bernie Sanders of Vermont,...