J. Scott Applewhite, File, Associated Press
FILE - In this Sept. 27, 2013 file photo, Sen. Ted Cruz, R-Texas, right, accompanied by Sen. Marco Rubio, R-Fla., speaks on Capitol Hill in Washington. Republican presidential rivals Rubio and Cruz are backpedaling furiously as they try to outmaneuver each other on immigration. Rubio co-wrote a massive 2013 immigration bill that passed the Senate. He disavows it now, but Cruz won’t stop talking about it.

WASHINGTON — Senate Majority Leader Mitch McConnell has informed Democrats he may schedule a vote next week on a hard-line immigration bill pushed by Texas Sen. Ted Cruz, a GOP candidate for president.

The move could bring debate over immigration to the Senate floor just as it flares on the presidential campaign trail. Cruz has been sparring with Florida Sen. Marco Rubio, another GOP hopeful, as front-runner Donald Trump pushes for millions to be deported.

The bill is called Kate's Law for a woman shot in San Francisco by an immigrant illegally in the country. It would increase penalties for re-entering the country illegally.

McConnell has been under conservative pressure to bring the legislation to a vote, particularly from Fox News Channel host Bill O'Reilly. But the move could give a platform to Cruz, with whom McConnell has sparred.

Democratic aides spoke anonymously Friday to disclose McConnell's potential plans. McConnell's spokesman, Don Stewart, said no decisions had been made.

The development came as Cruz and Rubio escalated their spat over immigration, each angling to appeal to conservative Republican primary voters.

Cruz went on conservative host Mike Gallagher's radio show to declare himself amazed at comments from Rubio suggesting the two senators hold similar stances on immigration. Rubio co-authored a comprehensive 2013 Senate bill that provided a path to citizenship to immigrants in the country illegally. Cruz opposed the bill, but he did support increasing legal immigration, one stance Rubio pointed to in making the comparison.

"That's like Obama saying my position is the same as his on Obamacare. That's like Ayatollah Khamenei saying my position is the same as his on the Iranian nuclear deal," Cruz said Friday. "It is laughingly, blazingly, on its face, false."

Rubio didn't back down, continuing to argue that Cruz, like him, supported a legal status for those in the country illegally. Rubio would go farther and allow citizenship, but either position is dismissed as "amnesty" by immigration hardliners.

"The point I was making is I'm surprised by his criticism given that fact that his record on immigration, quite frankly, is not substantially different than mine," Rubio said.

Cruz disputes that, saying an amendment where he proposed eliminating citizenship and allowing legal status for those here illegally was a tactic aimed at defeating the overall Senate legislation two years ago.