The Minnesota senator said the worries of "a few" Republicans comes from "the way that we all have this suspicion" that Trump "lies a lot."
"He says things that aren't true. That's the same as lying, I guess," he added, the Independent reported.
Franken cited Trump's evidence-free claim he would have won the popular vote in the presidential race if not for three to five million immigrants in the country illegally voting for his Democratic opponent Hillary Clinton.
Trump is also said to have told some senators in a private meeting at the White House that he and former Republican senator Kelly Ayotte would have won in New Hampshire if it had not been for voters bussed in from other states.
Franken said: "You know, that is not the norm for a president of the United States, or, actually, for a human being."
Last week, Franken first aired concerns in the Senate in an interview with Bill Maher: "Some will say that he is not right mentally. And then some are harsher."
He added: “That’s not fair. That was cheap. There are some who I guess don’t talk to me. I haven’t heard a lot of good things, and I’ve heard great concern about the president’s temperament.”
Franken also took issue with Trump reportedly telling a group of Democratic senators: "Pocahontas is now the face of your party."
The president's comments are a dig at senator Elizabeth Warren, who claimed Native American heritage.
He said he would have spoken up if present, saying: “Mr. President, with all due respect, that’s racist. Please stop doing that. I’m on [the] Indian Affairs [Committee].
"This is completely unacceptable. You really should stop doing this. It doesn’t serve anybody.”