King, a former chairman and current member of the House Intelligence Committee, was referring to news stories on Dec. 9 that cited either unnamed officials or intelligence sources and did not name Brennan as a source.
"There should be an investigation of what the Russians did, but also John Brennan and the hit job he seems to be orchestrating against the president-elect," Donald Trump, King said.
"I'm not saying it didn't happen," that Russia interfered in the election to help Trump. "I'm just saying we've been given no evidence on it, when we have these stories mysteriously appearing in the newspaper," King said.
A Democratic congressman, California Rep. Adam Schiff, his party's senior member on the House Intelligence Committee, pushed back against the criticism of Brennan over the information about the Russian hacking.
"He has done a remarkable job at the agency," Schiff said on "This Week." "I don't think he's trying to politicize this in any way."
Schiff continued, "I do think there's a consensus among the top leadership in the intelligence community, not only of who was responsible [for the hacking], but why. And I think when the review is released by the administration within the next two or three weeks, I think the American people will see that."
The California Democrat also cautioned that it is "deeply damaging" to the country for Trump to continue casting doubt on the capabilities of the intelligence community over Russia's election-related hacking.