Mitt Romney was already facing a hard time within his party when it comes to his stance on illegal immigration.

New Mexico Gov. Susana Martinez (R) (Susan Montoya Bryan/Associated Press)

Now, the presumptive GOP nominee faces criticism on the issue from a potential running mate, New Mexico Gov. Susana Martinez (R).

In an interview with Newsweek’s Andrew Romano, Martinez takes aim at “self-deportation,” an idea promoted by Romney during the GOP primary and one at the heart of Arizona’s controversial S.B. 1070 anti-illegal immigration law.

As we sit down at a local Starbucks, I ask about immigration. It’s a topic she has been reluctant to discuss since winning the Republican primary in 2010, so what comes next is surprising: a battle plan that contradicts nearly everything the GOP has been doing and saying since 2007, Romney’s “self-deportation” strategy included. “‘Self-deport?’ What the heck does that mean?” Martinez snaps. “I have no doubt Hispanics have been alienated during this campaign. But now there’s an opportunity for Gov. Romney to have a sincere conversation about what we can do and why.”

Martinez also suggests that the GOP propose its own comprehensive immigration reform measure in an effort to draw a contrast with the Obama administration’s lack of progress on the issue:

Naturally, Martinez has some suggestions. First, Republicans should remind Latinos that Obama pledged to pass comprehensive immigration reform by the end of his initial year in office, but “didn’t even have the courage to try.” Next, the GOP should outflank the president — on the left — by proposing its own comprehensive plan. “I absolutely advocate for comprehensive immigration reform,” Martinez says, sipping a caramel macchiato. “Republicans want to be tough and say, ‘Illegals, you’re gone.’ But the answer is a lot more complex than that.” Martinez envisions an approach “with multiple levels”: increased border security; deportation for criminals; a guest-worker program for people who want “to go freely back and forth across the border to work”; a DREAM Act-style pathway to citizenship, through the military or college, for children brought here illegally by their parents; and a visa (coupled with a “penalty” or a “tagback”) that allows the rest of the illegal population to remain in the U.S. while they follow standard naturalization procedures.

Romano notes that Martinez “also opposes a standalone DREAM Act, arguing that politicians can’t fix [immigration] by saying, ‘Here’s the DREAM Act and we’re done. It has to be part of a larger plan.’ ”

That last point would put Martinez at odds with another potential VP candidate, Sen. Marco Rubio (R-Fla.), who is working on his own scaled-back version of the DREAM Act and who argues that the issue of undocumented people who were brought to the United States as children is a “humanitarian” one.

A Romney spokeswoman did not immediately respond to a request for comment on Martinez’s remarks.

The interview highlights the difficulty for Romney of working to woo Latino voters while being ever mindful of a GOP base for which a hard-line stance on illegal immigration is of paramount importance.