Mr Johnson's decision to release the article just six days before Mrs May is due to set out her plans for Brexit in a speech in Florence prompted claims in the Mail on Sunday that allies of the PM believe the move is "hostile" and "attention seeking".
This was the claim that an additional £350 million could be spent on the NHS.
Johnson making his own views known first may be seen by many as a challenge to her authority, which has been weakened by a poor result in the June 8 election that cost her Conservative Party its majority in Parliament.
Home Secretary Amber Rudd.
"Both the United Kingdom and the EU would benefit from ensuring that no operational gaps are created" by Brexit, the statement from the Department for Exiting the European Union in London said. The Prime Minister must spell out now how this will be paid for, or stand condemned for once again trying to mislead the British public.
"This country will succeed in our new national enterprise, and will succeed mightily, " he said, dismissing any suggestion that Brexit will be reversed.
This comes as Mrs May is reportedly considering to pay up to £40billion in a Brexit divorce bill in order to keep access to the single market over a two-year transition.
Ms Rudd told BBC1's The Andrew Marr Show that she does not want Mr Johnson "managing the Brexit process".
Mrs Rudd said the Prime Minister is "driving the car".
Rudd said that while Johnson's article was "backseat driving" over Brexit, it was "absolutely fine" for him to intervene.
Will Tanner, a former adviser to the Prime Minister, tweeted it was "astonishing" that Mr Johnson thought "this self-serving posturing, disloyal at best of times, would reflect well in the hours after a terror attack".
4,000 words in the Daily Telegraph today are given to Boris Johnson to repeat the Vote Leave claim that Britain will still claw back £350m a week after leaving the European Union, with much of that money preferably being spent on the NHS.
"It would be very nice if we could get a clear message on where the British are" on their financial obligations with the EU, Danish Finance Minister Anders Samuelsen said in an interview in Copenhagen on Friday.
"Our systems of standards will remain absolutely flush with the rest of the European Union", he said, but when addressing the housing shortage he added: "There may be ways of simplifying planning procedures, post-Brexit, and abbreviating impact assessments, without in any way compromising the environment".