A Democratic lawmaker said Wednesday she hasn't had any luck in convincing President Barack Obama to visit the border between the U.S. and Mexico.

Second-term Rep. Gabrielle Giffords (Ariz.) said she hasn't been successful in luring the president or Attorney General Eric Holder to her district bordering Mexico, despite Holder's having filed a lawsuit Tuesday challenging Arizona's tough, new immigration law.

"You know, I haven't," Giffords said during an appearance on Fox News when asked if she had had any success in convincing Obama or Holder to visit.

Giffords is among a handful of centrist Democrats from border states and districts who, while opposing Arizona's immigration law, have discouraged the administration from challenging the law in court. Giffords and other Arizona Democrats have instead asked for more resources from the government.

"I certainly invite Attorney General Holder to come down and visit for himself, to see for himself what's going on in my district, southeast Arizona," she said. "I've repeatedly invited President Obama to come down."

Many administration officials have gone down to Arizona to tour the area, but Giffords dismissed those visits as meaningless.

"He certainly has sent some top-ranking officials," she said. "Unfortunately, those visits haven't been open to the public, so in essence, it's as if they're not even here."