The president also returned to his feud with the London mayor Sadiq Khan, who he previously sparred with via Twitter in the aftermath of the terrorist attacks on London in 2017.
The government will also seek mutual recognition of professional qualifications, it adds.
"I told Theresa May how to do it but she didn't agree, she didn't listen to me..."
While he added that he thought Johnson would one day be a good candidate to lead the country, he insisted that he wasn't trying to put Johnson against May.
"She wanted to go a different route".
Mr Coveney said: "What you will see from the European Union now is a 'take them seriously on this new position" and a desire to get the negotiating teams into a room from next Monday afterwards. And that is fine.
Mrs May faces a huge task trying to persuade European Union negotiators to accept that the proposals are viable, while also keeping her Conservative party and Parliament on side.
His comments went down badly in Downing Street as the Prime Minister desperately battles to contain a civil war within her party over her Brexit proposals.
The pro-sovereignty campaign, which advocates Britain exit the Single Market and its Free Movement regime cleanly, said in a statement: "This white paper is a complete capitulation and our government has become a national embarrassment. It was not the deal that was in the referendum".
Asked if the ex-minister could be in No. 10 one day, he replied: "Well I am not pitting one against the other. I don't know if that is what they voted for".
It wants to support businesses to move staff between the two territories, allows mobility for students and young people, and allow citizens to travel without visas for tourism and temporary business activity.
May has insisted that her Brexit plan will allow Britain to strike its own lucrative trade deals with countries like the USA after her country has left the bloc as planned in March next year.
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. "We are cracking down right now on the European Union because they have not treated the United States fairly on trading".
But Trump poured cold water on May's offer.
"If they do a deal like that, we would be dealing with the European Union instead of dealing with the United Kingdom, so it will probably kill the deal", Mr Trump told The Sun.
Good morning and welcome to our live coverage of today's politics news, which will be dominated today by Donald Trump's first official visit to the UK.
Outside the mansion, near Oxford to the northwest of London, a couple of thousand demonstrators lined the road and booed Trump's arrival.
"The President likes and respects Prime Minister May very much". He is scheduled to meet with Russian President Vladimir Putin in Finland on Monday.
But they avoided the noise and spectacle of the protests on the ground by travelling by helicopter between the airport, London and Oxfordshire.
An inflatable mockup of Trump in a nappy will also fly next to the British parliament.