It would also end free movement of people, and give the United Kingdom an independent trade policy and end annual payments into the European Union budget.
May's plan "will definitely affect trade with the United States, unfortunately in a negative way".
The UK outlined plans for a Ukraine-style association agreement with the European Union on Thursday (12 July) as it published its long-awaited White Paper on relations with the European Union after Brexit.
Trump came down firmly on the side of the Brexiteers, saying what May proposed would hurt the chances of a future trade deal between the UK and the United States.
"I am not pitting one against the other", the USA president said when asked if he'd like to see Johnson as PM.
He added that Mr Khan had shown a "lack of respect" for the American government by attacking him.
On Thursday, May invoked Winston Churchill as she addressed Trump and business leaders at a black-tie dinner at Blenheim Palace, the 18th century country house where Churchill was born.
Addressing the 100-strong group the Prime Minister said there was an "unprecedented" opportunity to do a deal that boosted jobs and growth in both countries.
In another blow to May, Trump said her now ex-foreign secretary "would be a great prime minister".
Tushar Arothe resigns as India Women's coach
In the meeting, BCCI acting Secretary Amitabh Chaudhary, GM (Cricket Operations) Saba Karim and CEO Rahul Johri were also present. However, it was twin defeat against minnows Bangladesh in the Asia Cup including one in the final, that became the last straw.
Asked about Johnson's comments at a private dinner two weeks ago that Trump "would go in bloody hard" if he was handling Brexit, the President said: "He is right". "I actually told Theresa May how to do it, but she didn't listen to me". "I'd like to see them be able to work it out so it could go quickly, whatever they work out". And that is fine. "But it is too bad what is going on", he told the tabloid, Britain's best-selling newspaper. "I don't know if that is what they voted for".
The government also insists, as expected, that it sees the EU's usual adequacy agreement - which the bloc has with 12 countries, in addition to two partial agreements - as "the right starting point" but that it should "go beyond the framework".
The government also appeared to recognise public concerns that Brexit negotiations need to start in earnest very soon in order for there to be no data "cliff edge", saying it was "ready to begin preliminary discussions on an adequacy assessment so that a data protection agreement is in place by the end of the implementation period at the latest". I have just been hearing this over the last three days.
Trump's visit comes at a time of political turmoil for the embattled British prime minister, who is fending off criticism over her plans for a "soft Brexit".
Trump arrived in Britain after a fraught North Atlantic Treaty Organisation summit in Brussels where he piled pressure on allies to double their defence spending.
Looking at the reception to May's paper from the Tory leader's own party, the Guardian reported that "almost all the reaction from Conservative MPs was either sceptical, or downright hostile", noting particular "alarm" being expressed over the role that the European Court of Justice (ECJ) looks set to continue playing in Britain's affairs post-Brexit.
As he began a four-day visit to Britain on Thursday, Trump, a major supporter of Brexit, told The Sun that he had advised May to go about leaving the European Union in a different way but was ignored.
But Mr Trump said the prime minister is a "very good person" denying claims he called her a bossy schoolteacher, or said she was boring.
Johnson isn't the only politician to have quit over May's Brexit plans, as ministers David Davis and Steve Baker also quit the government in protest.
The convention-shredding president had ignored diplomatic niceties ahead of touching down for the four-day visit - his first to Britain since taking power - by rebuking his beleaguered host as she struggles over Brexit.