WATCH: Ted Cruz dances around questions about Trump's wiretapping claim, takes on health care bill

Sen. Ted Cruz sidestepped questions about President Donald Trump's trustworthiness on "Face the Nation" Sunday, instead opting to hit the White House by denouncing its new health care bill.

To lead off the interview, "Face the Nation" host John Dickerson asked Cruz about Trump's allegation that his predecessor, former President Barack Obama, wiretapped his headquarters during the 2016 campaign.

"Well, these are serious allegations. I think they need to be looked into seriously," the Texas senator responded.

The Trump administration has failed to provide any evidence that Obama conducted surveillance on his campaign. Since Trump made the accusation, reports have come out indicating that the president learned of Obama's wiretapping from a right-wing radio host's baseless tirade. Still, Cruz and other Republican leaders have given credence to Trump's allegations.

Dickerson pressed Cruz on Trump's trustworthiness.

"You, in the campaign, went back and forth [with Trump] about the question of veracity," Dickerson said. "[Trump] told stories about your father and JFK's assassination. There seemed to be as much evidence for that as there is for this wiretapping claim. Can people trust this president?"

"I don't know what basis the president has for these allegations," Cruz said, answering a question nobody asked of him.

"This is not necessarily as outlandish as everyone in the press suggests," Cruz continued. "We do know that the Obama administration targeted their political enemies. We do know that that the IRS, for example, targeted citizens groups who spoke out in defense of the Constitution. The notion is not necessarily outlandish, but it is serious, so it needs to be based on facts."

Later in the interview, the Tea Party conservative blasted Trump's health care bill.

"I got to tell you, if Republicans hold a big press conference and pat ourselves on the back that we've repealed Obamacare and everyone's premiums keep going up, people are going to be ready to tar and feather us on the streets," he said. "And quite rightly."