(CNN) -- New Mexico Gov. Susana Martinez said Wednesday that she plans a new push to repeal the state law allowing undocumented immigrants to obtain driver's licenses.

Martinez, who has tried to get the law repealed twice before, described it as dangerous in a post on her official Facebook page.

"I am once again asking the legislature to repeal the law that gives driver's licenses to illegal immigrants," said Martinez, a Republican. "I am always willing to discuss this issue with legislators from both parties and explore ways to find common ground, but I believe the most effective solution is to simply repeal this dangerous law."

Her comments are the latest salvo in a nationwide debate over the controversial issue.

Some were hopeful that the nod toward compromise was a sign that her approach may have changed from past attempts to repeal the measure.

"We are encouraged that a compromise can be worked out by the legislators and the governor," said Allen Sanchez, executive director of the New Mexico Conference of Catholic Bishops, a group that has fought the governor's attempts to repeal the law.

Martinez, the first Latina governor in the United States, promoted the repeal of the 2003 law during her campaign for governor.

In 2011, she said the law had turned New Mexico into a magnet for people from other states seeking a license.

"We have created an industry of human trafficking," Martinez said in 2011. "People are getting paid to bring other people from all over the country to New Mexico for the whole purpose of defrauding our state."

New Mexico and Washington are the only two states that grant unrestricted driver's licenses to undocumented immigrants.

Utah issues driving privilege cards to undocumented immigrants who have lived in the state for more than six months. Illinois lawmakers approved a measure earlier this month to allow undocumented immigrants to get temporary licenses.

Several New Mexico lawmakers have said this month that they were considering a measure that would be a compromise more in line with the approach Utah and Illinois have taken, CNN affiliate KRQE reported.

"It doesn't benefit the citizens of the state of New Mexico for us to be at a stalemate." Democratic state Sen. Pete Campos told KRQE. "It is important for us to move forward."

For months, driver's licenses and other state benefits have been at the heart of a battle in the nationwide immigration debate.

In New Mexico, supporters of the licenses for undocumented immigrants argue that it's safer to have more drivers trained and insured, and opponents argue that it's a system rife with fraud.

Immigrant rights advocates have defended the current practice.

It was intended to offer undocumented immigrants "the opportunity to buy insurance, register their vehicles and have a driving record available to all law enforcement," said Marcela Diaz of Somos Un Pueblo Unido (We Are a United People).

But critics say the practice is turning New Mexico into a magnet for fraud.

"Currently, illegal immigrants from all over the country come to New Mexico to obtain a license without having the intention of staying here," Demesia Padilla, the state's secretary of taxation and revenue, said in 2011.

The state's Taxation and Revenue department has prosecuted a number of people, including Rosa Pardo-Marrufo, who confessed in 2009 to charging $700 to people who wanted to use her address in fraudulent applications.

CNN's Greg Morrison and Catherine E. Shoichet contributed to this report.