Yesterday Johnson's dad Stanley said he thought his son might be "happy to walk away from the whole thing" if his Brexit vision faltered.
She added: "This government is being run from the front and we all have the same destination in our sights".
Johnson published a 4,000-word plan for a "glorious" Brexit on Friday night, which included the claim that the United Kingdom would gain control of £350 million a week that could be spent on public services.
The former Tory leadership hopeful spoke to TV cameras in NY after bumping into them on his return from a jog.
Speaking in New York, Johnson was asked by reporters if he planned to resign in footage aired by Sky News.
May is will deliver a high profile speech on Brexit in Florence on Friday. "We are a nest of singing birds".
Behind the scenes, however, there has been significantly less harmony since Mr Johnson's unauthorised 4,200-word article in The Daily Telegraph on Saturday.
Hitting back at the foreign secretary, Sir David wrote in an article that he is "surprised and disappointed" by the repeat the figure of £350m per week, in connection with the amount that might be available for extra public spending when we leave the European Union.
That prompted a half-cent jump in the pound, to $1.3535 from around $1.3485 before the article was published.
According to the website "Boris Johnson "could not live with" the Swiss-style arrangement for Brexit being considered by Theresa May - and may even resign as Foreign Secretary before the weekend if the Prime Minister pushes ahead with such a plan, a key ally has told The Telegraph".
"Sounding off personally in this way is totally unhelpful and he shouldn't exploit the fact she hasn't got a majority in Parliament, and he knows perfectly well that normally the Foreign Secretary would be sacked for doing that - and she, unfortunately, after the general election, is not in the position easily to sack him - which he should stop exploiting".