WASHINGTON — When a visibly angry Senator Marco Rubio of Florida emerged as the face of hard-line Republican opposition to President Obama’s new outreach to Cuba, he swept political calculations aside.

“I don’t care if the polls say that 99 percent of people believe we should normalize relations in Cuba,” said Mr. Rubio, who grew up in Miami and is the son of Cuban immigrants.

Within the coming months, Mr. Rubio, a likely presidential candidate, will find out on just which side of the polls — and history — he stands. On Thursday, he got his first brush with an opposing view when Senator Rand Paul, Republican of Kentucky and a likely 2016 contender, endorsed Mr. Obama’s move to normalize relations with Cuba as “probably a good idea.”

The distance between their positions forms the contours of a debate within the party that will play out when Congress reconvenes under Republican control in January, and into the 2016 campaign. Jeb Bush, the former Republican governor of Florida, and Senator Ted Cruz, Republican of Texas, joined Mr. Rubio in criticizing the president. Gov. Chris Christie of New Jersey declined to comment.