Podesta is reportedly hesitant about taking the role, however.
"To all Americans, I see you & I hear you. I am your voice," Trump tweeted.
"Why are you even interviewing me?" he snapped.
The publication called Clinton the "most qualified presidential candidate in modern American history."
OPINION | Sanders, Democrats scrambling as millennial voters head toward third party
"They’re both incredibly supportive tax-and-spend liberals. ... Tax-and-spend doesn’t work."
"People are not happy. BIG lines. What is going on?"
Clinton's lead remains the same in a head-to-head pairing with Trump.
The current contract for "The Kelly File" host expires next year.
The push comes amid a new campaign ad calling Trump a "bully."