Tim Kaine, chairman of the Democratic National Committee, took a shot at Paul during a press conference in Washington Wednesday.
Kaine said Paul "represents the most extreme elements of the Republican Party - a candidate who has vowed to abolish the Departments of Education and the Federal Reserve; who vows to oppose, oppose, oppose at a time when we need constructive solutions to the challenges we face."
Asked about Kaine's comments in an interview set to air on CNN's John King, USA, Paul told Democrats to "bring it on."
The Kentucky Senate hopeful told CNN Chief National Correspondent John King that he thought having a budget deficit in excess of $1 trillion was "an extreme idea."
"You know, the debt is spiraling out of control and I'm proposing things like a balanced budget and they think that's an extreme idea? What I tell to the national Democrats is bring it on and please, please, please bring President Obama to Kentucky. We would want him to come and campaign for my opponent [Kentucky Attorney General Jack Conway]. In fact, we'll pay for his plane ticket if President Obama will come to Kentucky."
And Paul, the son of former Republican presidential candidate Rep. Ron Paul, also weighed in Arizona's upcoming Republican Senate primary. In that race, Sen. John McCain, the GOP's recent standard bearer, is locked in a tough contest with former Rep. J. D. Hayworth, who is also enjoying support from the Tea Party movement.
Asked whether he has a preference in the McCain-Hayworth race, Paul left the door open to backing Hayworth in his effort to unseat McCain.
"I haven't looked at it closely enough to know yet, but I do know that I have some similarities with former Congressman Hayworth and we will look at the race," Paul told King.
In the wide-ranging interview, the Kentucky Republican also touched on his positions on a number of hot-button policy issues; on his belief in smaller, limited government governed by the express terms of the Constitution; on term limits; on similarities between himself and his father; and on Senate Minority Leader Mitch McConnell, who backed Paul's primary opponent.
Watch the Paul interview on CNN's John King, USA Wednesday beginning at 7 p.m. ET.