WASHINGTON — As the specifics of immigration legislation take shape on Capitol Hill, President Obama is making it clear that he wants the overhaul on his terms.

Officials in the West Wing are convinced that the politics of the immigration issue have firmly shifted in their direction. That belief is fueling the president’s push for quick action and broad changes that go beyond what Republicans are signaling would be acceptable if they are to back legislation that allows a path to citizenship for millions of illegal immigrants.

The administration’s confidence — which was communicated to immigration advocates in a series of conference calls and meetings last week — is rooted in the sense among the president’s political advisers that Republicans are eager to embrace broad immigration changes as a way of improving their electoral appeal among Hispanic voters.

“We’re giving them some space,” said Dan Pfeiffer, a senior adviser to the president. But in the meantime, he said, “we’re going to continue to make the case to the country about why immigration reform should be done and to put pressure on Republicans that they need to do it.”