Shortly after the prime minister held talks with Trump at the White House on Friday, the new president signed an executive order to suspend refugee arrivals and impose tough new controls on travellers from seven Muslim countries.
Trump's move prompted an online petition to stop him making a planned state visit to Britain, a regal and glitzy affair which involves formalities such as a royal banquet in the Buckingham Palace Ballroom.
By late Sunday the petition to the British parliament had attracted over 800,000 signatures.
May sparked controversy Saturday after refusing to condemn Trump's immigration clampdown when pressed by journalists during a trip to Turkey, but later issued a stronger statement as it emerged British citizens had been affected.
"Immigration policy in the United States is a matter for the government of the United States, just the same as immigration policy for this country should be set by our government," a spokesman from her office said on Sunday.
"But we do not agree with this kind of approach and it is not one we will be taking," he added.
Foreign Minister Boris Johnson said it was "divisive and wrong to stigmatise because of nationality."
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UK wins exemption
High-profile British citizens caught up in the new US rules included double-double Olympic Champion Mo Farah, who slammed a policy based on "ignorance and prejudice" that could keep him apart from his US-based family.
"On January 1 this year, Her Majesty The Queen made me a Knight of the Realm. On January 27, President Donald Trump seems to have made me an alien," wrote the athlete, who represents Britain but was born in Somalia.
Iraqi-born MP Nadhim Zahawi, from May's Conservative Party, had earlier revealed he would be barred from entering the US under the clampdown.
"A sad, sad day to feel like a second class citizen! Sad day for the USA," he added.
But later on Sunday Johnson won an exemption for British citizens and dual nationals after he discussed the matter with Washington.
The foreign ministry subsequently announced the order would only apply to individuals travelling directly to the US from one of the seven listed countries -- Iraq, Iran, Libya, Somalia, Syria, Yemen and Sudan.
"If you are travelling to the US from anywhere other than one of those countries (for instance, the UK) the executive order does not apply to you and you will experience no extra checks regardless of your nationality or your place of birth," the foreign ministry said in a statement.
"If you are a UK national who happens to be travelling from one of those countries to the US, then the order does not apply to you -- even if you were born in one of those countries."
The foreign ministry added that dual nationals might have extra checks if they are coming from one of the seven named countries, "for example a UK-Libya dual national coming from Libya to the US".
Following the foreign ministry advice a spokeswoman for Farah said he was "relieved" but nonetheless "fundamentally disagrees" with Trump's order.
Zahawi praised the government for securing assurances for British citizens, adding that he still believed the new US rules amount to a "mistaken policy".
Pressure mounts on May
Trump's move prompted protests globally and demonstrators are due to gather outside Downing Street on Monday and hold rallies in cities across the UK.
Domestic pressure has mounted on May to distance herself from Trump, with opposition Labour party leader Jeremy Corbyn urging the government to put off the president's scheduled state visit until the ban is rescinded.
May "would be failing the British people if she does not postpone the state visit & condemn Trump's actions in the clearest terms," he wrote on Twitter.
The online petition against Trump's planned state visit later this year will be considered for debate in parliament, but a government spokesman said the invitation "was extended and has been accepted" by Washington.
