It is expected that the prime minister will deal with the issue in questions after her speech, which will address the UKs involvement in Middle East security, including training for Iraqi forces to clear Mosul and Raqqa of explosive devices left by Isis as it fled, and more security assistance for Jordan.
Among the likely questions will be whether she regrets being the first overseas leader to visit Trump after his inauguration, and his planned state visit to the UK.
Asked if she was shocked that Trump was directly attacking a close ally, Greening told the BBCs Today programme: The UK and US have been longstanding allies and our relationship with America is a hugely important one, and I dont think we should allow this tweet to undermine that in any way.
[Our relationship] will succeed long after presidents come and go I dont agree with the tweet President Trump has made but I also dont believe it should distract from the agenda we have domestically or detract from the close relationship the UK has had for many, many years and will go on to have with the American people.
However, Sajid Javid, the local government secretary, who is Muslim, took a much harder line. He posted on Twitter: So POTUS has endorsed the views of a vile, hate-filled racist organisation that hates me and people like me. He is wrong and I refuse to let it go and say nothing.
The mayor of London, Sadiq Khan, who has previously clashed with Trump, also issued a strongly-worded statement of condemnation, calling on the prime minister to demand an apology.
The feud marks a new, unexpected twist in the special relationship that has benefited from personal chemistry between leaders such as Winston Churchill and Franklin Roosevelt, Margaret Thatcher and Ronald Reagan, and Tony Blair and Bill Clinton.
There had been hopes that May and Trump whose mother was British would achieve a similar rapport. She was the first foreign leader to visit after he took office, they were photographed holding hands at the White House and she invited him on a state visit to the UK. But that has yet to take place after a series of controversies and warnings that protesters will take to the streets to show he is not welcome.
Hostility in the UK deepened on Wednesday when Trump highlighted videos from the feed of Jayda Fransen of Britain First that purported to show a group of Muslims pushing a boy off a roof. Another claimed to show a Muslim destroying a statue of the Virgin Mary, and a third claimed to show a Muslim immigrant hitting a Dutch boy on crutches.
The credibility of the last video was immediately undermined when the the Dutch embassy in the US said the perpetrator of the violent act in the video was born and raised in the Netherlands. Fransen has been charged with using threatening or abusive language following an appearance at a far-right rally in Belfast this summer.
Mays spokesman made clear Trumps invitation still stood but said it was wrong for the president to have done this. Boris Johnson, the foreign secretary, added: UK has a proud history as an open, tolerant society & hate speech has no place here.
The Labour party leader, Jeremy Corbyn, described the retweets as abhorrent, dangerous and a threat to our society. He and several other members of parliament called for the state visit to be cancelled.
The Labour MP David Lammy posted: Trump sharing Britain First. Let that sink in. The President of the United States is promoting a fascist, racist, extremist hate group whose leaders have been arrested and convicted. He is no ally or friend of ours. @realDonaldTrump you are not welcome in my country and my city.
Justin Welby, the archbishop of Canterbury, urged Trump to remove the retweets. And Brendan Cox, widower of Jo Cox, an MP murdered last year by a man reportedly shouting Britain first as he shot and stabbed her, told CNN: I think we probably got used to a degree of absurdity, of outrageous retweets and tweets from the president, but I think this felt like it was a different order.
Here he was retweeting a felon, somebody who was convicted of religiously aggravated harassment, of an organisation that is a hate-driven organisation on the extreme fringes of the far, far right of British politics. This is like the president retweeting the Ku Klux Klan.
US Democrats joined the condemnation. Keith Ellison, the deputy chair of the Democratic National Committee and a Muslim member of Congress, branded the president a racist.
But the White House defended the retweets. The principal deputy press secretary, Raj Shah, told reporters on Air Force One: We think that its never the wrong time to talk about security and public safety for the American people. Those are the issues he was raising with the tweets this morning.
Asked if Trump was aware of the source of the tweets, Shah replied: I havent spoken to him about that.
The spokesman insisted: The president has the greatest respect for the British people and for Prime Minister May.
Trumps new salvo echoed his criticism in June of Londons mayor, Sadiq Khan, after seven people were killed and 48 injured in a terror attack in the city. Khan, the first Muslim mayor of a western European capital city, hit back on that occasion and tweeted on Wednesday: Britain First is a vile, hate-fuelled organisation whose views should be condemned, not amplified.