The Foreign Secretary reacted to articles in the Daily Telegraph and The Sun claiming Mr Johnson was calling for Chancellor Philip Hammond to be sacked and would refuse to leave his post if he was demoted by Theresa May.
In a WhatsApp message to Conservative MPs, he wrote: “I am frankly fed up to the back teeth with all this.
“I do not know who these people are. I do not know if they are really my friends and allies or if they represent some sinister band of imposters.
“I heartily disagree with the sense, tone and spirit of what they are quoted as saying.
“Whoever they are they do not speak for me.”
Mr Johnson’s message comes as Mrs May prepares to brief Parliament over the status of Brexit negotiations and as talks resume in Brussels.
The Prime Minister has come under pressure to sack both Mr Johnson and Mr Hammond as she tries to reassert her authority over the Tory party.
Mrs May vowed she would not “hide from a challenge” amid speculation she could reshuffle her Cabinet to shore up her precarious position.
On Monday, The Daily Telegraph reported that Mr Johnson would resist any attempt to demote him, forcing Mrs May to either leave him in post or sack him.
Mr Johnson has been accused of undermining the PM by putting forward his own vision for Brexit.
A spokesman for the Prime Minister said on Monday that Mrs May has full confidence in Mr Hammond and Mr Johnson.
He said: “I never discuss whether reshuffles are or are not taking place. But as the Prime Minister said yesterday in the Sunday Times, she has a fantastic Cabinet.”
Mrs May has also faced repeated questions on whether Mr Johnson is “unsackable” after she lost her Commons majority in the general election.
Last week, Mr Johnson called on Conservative colleagues to move on from “malarkey” surrounding Mrs May’s position and to “get behind the PM” following her difficult party conference speech.
He wrote: “Get behind the PM. Ordinary punters I have spoken to thought her speech was good and anyone can have a cold.”
It comes after it emerged that Welwyn Hatfield MP Grant Shapps claimed to have the backing of around 30 MPs, including “one or two” Cabinet ministers, calling for Mrs May to resign.