The president had tipped Mr Johnson for the job when he said the businessman was “going to St James’s” during a luncheon in January.
At the time the appointment was welcomed as a “stylish” choice as the US’s top diplomat in London, also known as ambassador to the Court of St James’s.
Mr Johnson, 70, helped raise money for Mr Trump’s presidential campaign and donated funds to help pay for his inauguration.
The billionaire businessman and philanthropist from the Johnson & Johnson family is chairman and chief executive of the Jets, although will hand over to his brother, Christopher, should the nomination be approved.
The team’s president, Neil Glat, said: “On behalf of the entire organisation, we would be extremely proud to see Woody nominated and confirmed as US ambassador. We could not be more excited for him and his family.”
The nomination has to be approved by the US Senate, which will also decide on Mr Trump’s choice of ambassador to Belgium, Jamie McCourt.
Ms McCourt was co-owner and chief executive of the Los Angeles Dodgers baseball team until 2009.