It’s entirely possible. NBC is not (sadly) a legit news source these days. (This is the parent company of MSNBC after all, and the network that still employs Brian Williams for big bucks.)
And though NBC isn’t trustworthy, it’s not hard to imagine Trump saying what was reported.
For the record, we don’t trust (almost) anyone in Washington. That is a good rule of thumb for pretty much everyone. People in DC live by it. (It’s the only way to survive.)
Also the FCC shouldn’t even exist.