Sending a direct warning to the EU, Mr Gill said: "If the EU then demanded we couldn't begin negotiations with other countries we should refuse to apply the EU's tariffs for them on products ultimately destined for the continent".
But many Brexit supporters have cast May's negotiation as a betrayal, including lawmakers in her own deeply divided party, who have warned May might face a leadership challenge. "But it is too bad what is going on", he told the tabloid, Britain's best-selling newspaper.
Liam Fox, Britain's trade minister, tweeted after the Sun interview was published that it was terrific to hear Trump "talk so positively about UK & US Trade tonight at Blenheim Palace".
The Sun said its interview with Trump was conducted in Brussels, where the North Atlantic Treaty Organisation summit was held earlier in the week. It was one of more than a hundred protests police expected during Trump's four-day trip.
The EU has consistently made it clear there can be no "cherry picking" of the single market - and that's precisely what this document does.
Hammered out at Theresa May's Chequers retreat, the plan has upset Tory Leavers, who fear it doesn't amount to a "hard Brexit", with David Davis and Boris Johnson quitting their ministerial posts in protest.
Trump accused the British political leader of "killing" chances of a USA trade deal by ignoring his advice on Brexit and taking her own, "softer" approach.
"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", Trump said, adding that, "We have enough difficulty with the European Union".
"I actually told Theresa May how to do it, but she didn't listen to me".
In fact, much of Britain's division over Brexit - which has split the governing Conservative party and the public at large - stems from the June 2016 referendum on withdrawing from the European Union not including language about would come next.
Prime Minister Theresa May's government is trying to present plans that satisfy Britons who voted for their country to leave the bloc and set an independent course, but without hobbling businesses, security agencies and other sectors that are closely entwined with the EU.
Ireland's deputy PM Simon Coveney said the European Union would now take "seriously" the UK's proposals. "That is exactly what we will do".
He also added that Johnson, "would be a great Prime Minister".
"You know I've been reading a lot on Brexit over the last couple of days and its seems to be turning a little bit differently where they're getting at least partially involved back with the European Union", he said.
Trump blamed London mayor Sadiq Khan for allowing the protests and blasted the city's first Muslim leader as weak on terror and crime.
However, Labour leader Jeremy Corbyn said the president's policies were putting millions of lives at risk and criticised May for her warm welcome.
Trump did not exactly give May's plans a ringing endorsement.
Labor officials are not talking about their tactics for the coming battles but, according to Politico, have spoken privately about a "coalition of chaos" where their MPs would vote with Brexit hardliners to bring down May's deal in an attempt to destabilize the government and force a new election.
May's critics have said, however, that she reins in criticism of Trump because of the strong desire of many in her Conservative Party to reach a trade deal with the United States after Brexit.
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".
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