The Trump administration last week formally asked the Supreme Court, the highest court in the US, to allow the ban to take effect, arguing that restricting immigration by refugees and visitors from Iran, Libya, Somalia, Sudan, Syria and Yemen is necessary to protect USA national security.
"I don't know how to tell you this, but I really don't care", he told The Associated Press.
Prime Minister Theresa May, asked by The Sun newspaper whether the controversial trip would go ahead, answered "Yes". "If he does a good job and frankly if he does a great job, that would be a terrific thing".
Trump congratulated Khan on his victory, stating he was "happy to see that". "It's not about me". That's not what I said.
It was not until the Westminster attack in March of this year that Mr Khan again had friction with the Trump family.
President Donald Trump's two adult sons are defending their father's controversial Twitter attack against London Mayor Sadiq Khan.
Trump Jnr, an executive of the Trump Organisation, said: "You have to be kidding me?!" It also was unclear whether he was aware the six-month-old piece bore no relation to the attack outside the Houses of Parliament and instead centred on Mr Khan's reaction to a bombing in the Chelsea neighbourhood of NY.
A spokesperson for Khan on Sunday called Trump's remark "ill-informed" and said it took the mayor's words out of context.
In a series of tweets, President Trump challenged Mr Khan for saying there was "no reason to be alarmed".
Johnson said separately: "He is entirely right to say what he said to reassure people of his city about the presence of armed officers on the streets".
The latest opposition comes after Trump sought to ridicule the response of London mayor Sadiq Khan to last week's terror attack, in which seven people were killed. "MSM is working hard to sell it!"
Many in the United Kingdom have been angered by the apparent aggression against the London Mayor, just as his city was facing dark times.
On Monday, Khan said, "I don't think we should roll out the red carpet to the president of the U.S.A.in the circumstances where his policies go against everything we stand for". Dan Gross of the Brady Campaign highlighted Trump's tweet while responding to the Orlando workplace shooting on Monday that left at least five people dead, which he called a "tragic example of the American gun violence that kills 93 people every day".