As President Trump continues to slam special counsel Robert Mueller's probe into Russian interference in the 2016 election, Texas Republican Sen. John Cornyn says that Mr. Trump appears to be "conflating" that aspect of Mueller's probe with questions about possible collusion between the Russian government and members of the Trump campaign. He told "Face the Nation" that Mr. Trump takes the investigation personally and "feels like he's been under attack."

"The politics of this are very tempting to our friends across the aisle. But I think he's conflating those two, as are a number of other people, and we need to keep those separate. Yes, the Russians did meddle. Number two, did any American involved collude with them? And so far there's no evidence," Cornyn said Sunday.

Cornyn's comments come after the Justice Department announced a new indictment against 12 Russians for their alleged involvement in hacking Democratic computer systems and stealing party data. In an interview with "CBS Evening News" anchor Jeff Glor on Saturday, Mr. Trump said that he "hadn't thought of" asking Putin to turn over the accused Russians during the leaders' meeting in Helsinki, Finland, on Monday.

"I hadn't thought of that. But I certainly, I'll be asking about it. But again, this was during the Obama administration. They were doing whatever it was during the Obama administration," Mr. Trump said.

Cornyn, meanwhile, said that Americans need to "wake up" to Russia's efforts to "undermine confidence in our democracy."

"This is a warning shot for our next elections and we'd better be ready," Cornyn added. "I agree that the Russians did meddle in our elections and we better be prepared for the next time. But again there's no indication that it had an outcome -- that it changed outcome of the election."

Despite widespread concerns from Republicans and Democrats over the president meeting privately with Putin, Cornyn said that "it's always helpful when leaders of nations talk to one another."

"To me it would be counterproductive just to ignore that or to avoid that possibility. On the other hand, I think the president should be clear-eyed about who he's dealing with. Putin is an autocrat. He's a thug. He does not respect the rule of law. Obviously doesn't respect our democracy and wants to undermine it every chance he gets. But on the other hand there are commonalities here in terms of our desire to fight Islamic extremism around the world," Cornyn added.