Northern Ireland Secretary Chris Heaton-Harris said someone has spread a fake email claiming he was resigning.
Mr Heaton-Harris clarified that he has not resigned from his role after the email was circulated.
The fake statement was emailed to print titles on the Northern Ireland Office (NIO) circulation list on Wednesday evening.
It comes after the Northern Ireland Secretary extended the deadline for calling a fresh Stormont election to January 19.
Mr Heaton-Harris said: “Hello Twitter! Someone has sent a fake e-mail to press outlets saying I’ve resigned.
“This is totally untrue. I hope one of @elonmusk first moves is to eliminate fake news on Twitter… Very exciting I know, but complete and utter tosh.”
The email included fake quotes from the Northern Ireland Secretary citing reasons for his alleged resignation.
A short time later a legitimate email was sent by the NIO with the subject line “Fake email this evening”.
A spokeswoman said an email had been sent with a “fake quote from the Secretary of State”.
She asked members of the media to “disregard this email which is not a verified Government account”.
Mr Musk responded directly to Mr Heaton-Harris on Thursday morning.
Replying to the tweet in which he had been tagged, the Twitter chief executive posted: “What does a tosh look like?”