Hillary Rodham Clinton recently spent four days straight scurrying across the Western United States, mingling for hours at a time with hundreds of Democratic donors from the Rockies to Portland, Ore., from Southern California to the southern tip of Texas, and everywhere seeking the maximum primary campaign contribution allowed by law: $2,700.

But her breakneck pace, and the kind of old-fashioned face-to-face fund-raising that Mrs. Clinton and her husband, former President Bill Clinton, have perfected over the years, are no match so far for the cash pouring into Republican “super PACs” and other groups with no donation limits, one gargantuan check at a time.

Republican presidential candidates have gained a near monopoly on donors of $1 million or more, a New York Times analysis of financial records shows. Fifty-six donors gave at least that much to committees supporting Republicans like Senator Ted Cruz of Texas, former Gov. Jeb Bush of Florida and former Gov. Rick Perry of Texas, for a total of $124.2 million — about 12 times as much as Democrats’ biggest donors have given.

A single Republican contributor — Robert Mercer, a hedge fund magnate who gave $11.3 million, nearly all of it to help Mr. Cruz — surpassed all of the million-dollar donors supporting Mrs. Clinton, combined.