Republican presidential candidate Jeb Bush on Sunday doubled down on his attack against his 2016 rival Marco Rubio for his record of missing Senate votes, which Bush says is a sign that Rubio has "given up" on Congress.

Bush first criticized Rubio about his Senate record during the third Republican presidential primary debate last Wednesday, when he called him out for failing to appear and vote as an elected U.S. senator of Florida. The former Florida governor also suggested that Rubio resign from his seat in order to continue campaigning.

Bush then reiterated his criticism of the freshman senator over the weekend while speaking to NBC's "Meet The Press," questioning Rubio's leadership skills and drive to break through political gridlock in Washington.

"The basic point with Marco isn't that he's not a good person or he's not a gifted politician; everybody can see that. It's that I have proven leadership skills," Bush said. "I got to be governor of a state and accomplish big things. And in this era of gridlock, it's really hard to break through, and I think he's given up. And I think that's the wrong thing to do."

"This is about public service, about solving problems. I can change the culture in Washington," the former two-term governor added.

Rubio responded to Bush's comments Sunday on CBS's "Face the Nation," saying his focus on beating Democratic front-runner Hillary Clinton was more important than the meaningless Senate votes he missed.

"The truth is, I don't like missing votes, but what I would really hate is to wake up on the first Wednesday of November to the news that Hillary Clinton's been elected president," Rubio said, according to CNN. "Far too many votes today in the Senate are predetermined. We know what outcome's going to be. It's being done for messaging purposes, but it's never going to pass."

Rubio added that he does not plan to attack Bush and instead praised his leadership as governor.

"You're never going to hear me badmouth him. He was a great governor of Florida. He's someone I have tremendous personal affection for," he said.