LAS VEGAS — Seizing an opening to rewrite the nation’s immigration laws, President Obama challenged Congress on Tuesday to act swiftly to put 11 million illegal immigrants living in the United States on a clear path to citizenship.

But his push for speedy action and his silence on proposals to defer the opportunity for legal residency until the borders are deemed secure provoked criticism from a Republican leader on the issue. The response suggests that reaching consensus on immigration law changes remain difficult despite a new bipartisan push since the November elections.