In a sense, Mr. Schumer’s incarnation as a conservative bogyman resembles the role Edward M. Kennedy, the longtime liberal senator from Massachusetts, played in previous Republican primary campaigns. In the 2008 presidential race, conservatives attacked Senator John McCain, Republican of Arizona, for helping write the McCain-Kennedy bill overhauling the immigration system. (Mr. Kennedy died in 2009.)

In 2015, Republican Senator Ted Cruz of Texas bragged of having opposed “Chuck Schumer’s Gang of Eight amnesty legislation.” Mr. Cruz and Senator Rand Paul, Republican of Kentucky, have criticized the Rubio-Schumer bill, and influential conservative radio hosts like Rush Limbaugh and Laura Ingraham have homed in on Mr. Schumer’s role in designing the legislation.

Backers of Mr. Cruz have paid for radio ads that credit him for fighting hard in opposition “when Chuck Schumer and Marco Rubio tried to push amnesty.” Ms. Conway, who is advising one of the outside groups that supports Mr. Cruz, said a pro-Cruz “super PAC” would soon air television commercials tying Mr. Rubio to Mr. Schumer.

At a recent Rubio event in Iowa, an unknown antagonist circulated fliers branding him as “Chuck Schumer’s amnesty pitchman.” And at the most recent Republican presidential debate, Mr. Schumer’s name came up four times. (Ms. Pelosi, whose name was once instant anathema to Republicans, went unmentioned.)