Republican presidential candidates Marco Rubio and Ted Cruz continue to battle on who has the strongest stance on legal immigration.

Following their heated exchange on last Tuesday's Republican presidential debate, where Cruz said Rubio favored "amnesty" with President Barack Obama and Sen. Chuck Schumer, D-N.Y., the Florida senator appeared on Fox News to further debunk the Texas senator's claims and turn the tables around.

"[Cruz] clearly was a supporter of legalization. In the amendment, he talked about how he wanted the bill to pass," said Rubio on Wednesday. "He said I want immigration reform to pass. Basically, the argument he was making is we can pass immigration reform but we can't do citizenship. Let's just do legalization, and if we do that this bill has a chance of passing."

Although Rubio says he personally likes Cruz, the former acknowledged they have their differences.

Rubio explained his support for comprehensive immigration reform in 2013. At the time, he was a co-sponsor of the "Border Security, Economic Opportunity, and Immigration Modernization Act" (S. 744), but he has been criticized for drifting away from the bill. S. 744 did pass the Senate but it was never picked up for debate in the House of Representatives.

"On the issue of immigration ... I live in a state that is deeply impacted by immigration and I said I want to try and make a difference. Now, I know that [then-Senate Majority Leader] Harry Reid controls the senate, so let's do the best we can in the Senate. Hopefully the House can make it even better and let's see if we can make some progress," Rubio said, adding that he now realized nothing can get accomplished until illegal immigration levels are under control.

Cruz said his stance includes opposition to both a pathway to citizenship and a pathway to legalization. With the S. 744 bill, Cruz submitted an amendment that would deny immigrant citizenship but would provide green cards and work permits.

Cruz was confronted with his own words during a May 2013 speech, when he said he does not want immigration reform to fail and wants a bill "that allows those that are here illegally to come in out of the shadows." Cruz reiterated that he submitted an amendment, essentially a "poison pill," that would remove the pathway to citizenship provision.

"The fact that I introduced an amendment to remove part of the Gang of Eight bill doesn't mean I support the rest of the Gang of Eight bill," said Cruz on Fox News on Wednesday.

On June 27, 2013, S. 744 received 68 votes in favor and 32 against. Voting in favor, with Rubio, were fellow Republican presidential candidate Lindsey Graham, 2008 Republican presidential candidate John McCain and current Democratic presidential candidate Bernie Sanders. Fellow Latino Sen. Bob Menendez, D-N.J., also voted in favor of the bill.

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