WASHINGTON — Two days after taking veiled jabs at his rival, Sen. Ted Cruz took off the gloves and called out Sen. Marco Rubio by name Thursday for his role in helping to author a 2013 immigration bill that would have provided immigrants in the U.S. illegally with a potential path to citizenship.

Cruz’s comments, on a program with conservative radio host Laura Ingraham, set off a war of words between the two senators, both children of immigrants, with each arguing the other was soft on illegal immigration. And that is raising concerns among some party strategists that the high-profile fight could further alienate Latino and Asian-American voters, wrecking the party’s chances in a general election where 30 percent of the electorate is projected to be nonwhite.

“This is disaster on all kinds of different levels,” said John Feehery, a veteran Republican strategist and lobbyist. “I’ve always been concerned that if we don’t get immigration right we have no chance to win this. And right now it doesn’t look like we are getting it right or we’re going to get it right.”

“Talk is cheap. You know where someone is based on their actions,” Cruz told Ingraham.

In South Carolina, Rubio fired back that his Texas rival’s past positions weren’t “dramatically different” than his, citing amendments Cruz proposed to the 2013 bill to keep the work permits for undocumented people (while stripping out the citizenship component) and to increase H-1B work visas for highly skilled foreign workers.

The showdown comes as Rubio and Cruz rise in the polls while picking up endorsements and donors, fueling predictions that the fight for the Republican nomination could come down to a duel between the two young Cuban-American senators. Republican strategists are nervous that a race to the right on immigration could sink them in the general election.

Katie Packer Gage, the deputy campaign manager for 2012 presidential nominee Mitt Romney, who lost an election in which President Barack Obama was overwhelmingly supported by Hispanic voters, urged Cruz and other Republicans not to alienate more Hispanic voters in 2016.