In Washington, Republicans keep taking steps that imperil their relationship with Hispanic voters, passing legislation to accelerate the deportation of Central American children at the southern border and comparing their influx to a warlike “invasion,” compounding an electoral disadvantage that many in the party are convinced cost them the White House in 2012.

But in a vivid display of the strife within the party over immigration, likely contenders for the 2016 Republican presidential nomination are charting their own, very different course: delivering pointed and personal overtures to the native countries from which millions of Hispanic Americans have immigrated.

The result is a tug of war between congressional Republicans, who are unleashing ever-harsher language and legislation to rein in illegal immigration, and party leaders with their eyes on the White House, who are determined to build bridges to a crucial constituency that has long been neglected.