New Jersey Gov. Chris Christie (R). (John Gress/ Reuters)

One key line of attack against the Republican party's top presidential contender, New Jersey Gov. Chris Christie, revolves around his conservative bona fides: Is he or isn't he?

Democratic officials want to paint him as a right-wing, hard-line conservative. But Christie's potential rivals want to slap him with a different label: moderate. Kentucky Sen. Rand Paul, another GOP contender, used that word to describe Christie this week, adding that it would be difficult for a moderate to carry key conservative states. So which does Christie prefer? Neither, he says.

"All the labels, that's for the folks down in Washington, D.C.," Christie said on CBS's "Face the Nation" on Sunday morning, repeating a response he provided to similar questions on other shows. "And obviously they love playing that game, but the people of America aren't interested in that game, and I think given the approval ratings in Washington, they've shown that."

Christie's handful of appearances on the Sunday shows were each underscored by a question over his conservative credentials. On immigration reform, which many conservatives oppose, Christie repeated his assertion that the "broken system" needs to be fixed, though he didn't specify what was broken or how to fix it. On gun control, he stressed that he was focused on policies that address mental illness, noting that most mass shootings were perpetrated by individuals with histories of mental illness.

The New Jersey governor repeatedly pushed back against questions over whether he could lead a party that's been pulled further to the right in recent years.

"I'll work with anyone and everyone who is willing to work with me consistent with my principles," he said on "Fox News Sunday."