He tweeted that Mr Khan - who had earlier said the UK should "not roll out the red carpet" for Mr Trump - was a "stone cold loser", but the president added he was looking forward to his visit.
Protests are planned in several UK cities during the three-day visit, including London, Manchester, Belfast, and Birmingham.
Talks between Mr Trump and outgoing Prime Minister Theresa May will begin on Tuesday, with the pair expected to discuss climate change and Chinese technology firm Huawei.
Crowds were gathered outside Buckingham Palace as the president and first lady landed by helicopter shortly after midday.
After a private lunch, the couple will be given a tour of Westminster Abbey and will also meet Prince Charles and the Duchess of Cornwall for tea at Clarence House.
Later, the Queen will host a state banquet at Buckingham Palace, also attended by the Duke and Duchess of Cambridge.
The Duchess of Sussex will not attend following the birth of her son Archie, who is less than a month old. On Sunday, Mr Trump denied calling the duchess "nasty", despite him using the word on tape.
As he stepped onto UK soil, Mr Trump was greeted by US Ambassador to the UK Woody Johnson and Foreign Secretary Jeremy Hunt.
Tory leadership candidate Mr Hunt, who has spoken about the importance of the UK's relationship with the US, said Mr Trump mentioned to him "some of his very strong views about the Mayor of London" which he had also tweeted.
Mr Trump's tweet accused Mr Khan of doing a "terrible job" as mayor, adding: "[He] has been foolishly "nasty" to the visiting president of the United States, by far the most important ally of the United Kingdom. He is a stone cold loser who should focus on crime in London, not me."
A spokesman for Mr Khan said "childish insults" should be beneath the US president, adding: "Sadiq is representing the progressive values of London and our country, warning that Donald Trump is the most egregious example of a growing far-right threat around the globe."
Although Mr Trump has spoken of his admiration for Mrs May, there is expected to be differences of opinion during their talks, which begin on Tuesday.
The prime minister will raise the issue of climate change, with a government spokesman again saying on Monday the UK was "disappointed by the US decision to withdraw from the Paris Agreement in 2017".
Before the visit, President Trump told the Sun newspaper he was backing Conservative Party leadership contender Boris Johnson to be the next UK prime minister.
He also told the Sunday Times that Brexit Party leader Nigel Farage - an arch critic of Mrs May - should be involved in the government's negotiations to leave the EU.
At Monday evening's Buckingham Palace banquet, both the president and the Queen are expected to make a speech to guests, who will include prominent Americans living in Britain.
Labour leader Jeremy Corbyn, House of Commons Speaker John Bercow, and Liberal Democrat leader Sir Vince Cable are all boycotting the state banquet.
Hundreds await palace arrival
By Gary O'Donoghue, BBC North America correspondent
Hundreds gathered outside Buckingham Palace to glimpse the arrival of Marine One bringing the president and the first lady for lunch with the Queen and other members of the "firm".
All morning we've had a steady stream of pundits and politicians, journalists and royal watchers clambering their way over the cables to talk to broadcasters from all around the world.
At one point, Lib Dem leader Sir Vince Cable is challenged on his decision not to attend tonight's banquet.
Afterwards, he tells me he'll probably have dinner at home instead: "Maybe fishfingers," he says, chuckling.
As well as the great and the good, the police sniffer dogs have been here too. A busy cute little thing did a good deal of panting around our bag of sandwiches and crisps.
"He quite likes chocolate," said his female handler.
I cannot help wondering what he would make of the lunch about to be served across the road.
Mr Trump's daughter, Ivanka, arrived in the UK before the president. On Sunday, she posted a picture from outside the Victoria and Albert Museum in west London where she had visited the Christian Dior exhibition.
On Tuesday morning, President Trump and Mrs May will host a business breakfast at St James's Palace.
Business leaders expected to attend include Barclays CEO Jes Staley, GlaxoSmithKline chief executive Emma Walmsley, BAE Systems chairman Sir Roger Carr and the National Grid's John Pettigrew.
President Trump will then visit Downing Street for further talks with Mrs May, followed by a joint press conference.
Protesters are organising a "national demonstration" which will start at Trafalgar Square at 11:00 on Tuesday.
Both the Stop Trump Coalition and Stand Up to Trump protest groups said they would be present.
The Met Police said it had "a very experienced command team" leading the operation to deal with the visit.
On Wednesday, President Trump will visit Portsmouth to mark the 75th anniversary of the D-Day landings.
He will then fly to Shannon, in Ireland, to meet Irish Taoiseach (Prime Minister) Leo Varadkar, before going to his golf course at Doonbeg in County Clare.
Although the Queen has met 12 of the 13 US presidents who have been in office during her reign, Mr Trump's state visit to the UK is only the third by a US leader.
George W Bush and Barack Obama are the only other US presidents to be given a state visit.
State visits differ from official visits and are normally at the invitation of the Queen, who acts on advice from the government. The Queen usually receives one or two heads of state per year and has hosted 112 of these visits since becoming monarch in 1952.
