May had said the immigration policy was a matter for the US, before Downing Street today clarified she "does not agree" with it. Her initial comments and refusal to condemn Trump's executive order on the ban had sparked controversy yesterday.
"Immigration policy in the United States is a matter for the government of the United States. But we do not agree with this kind of approach and it is not one we will be taking," a Downing Street spokesperson said.
She was, by then, in Turkey, where she declined to condemn the order before Downing Street clarified her stand today.
Protesters are planning to assemble outside Downing Street to criticise May's response and a petition calling for Trump's proposed state visit to the UK to be cancelled crossed over 100,000 signatures, prompting a possible House of Commons debate.
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There are fears that British athletes, including Somalia-born Olympic champion Mo Farah, who trains in the US, could be affected by the ban.
Labour leader Jeremy Corbyn accused her of a "weak failure" to stand up for British values, indicating that Trump should not be invited to the UK for a state visit intended for June.
May had invited Trump to Britain on behalf of Queen Elizabeth II when she visited him in Washington DC earlier this week and the US President had accepted her invitation.
"Donald Trump should not be welcomed to Britain while he abuses our shared values with his shameful Muslim ban and attacks on refugees' and women's rights," he said.
Liberal Democrat leader Tim Farron told BBC: "She should've argued against this from the beginning, but once it became apparent that this was going to affect British people. You would expect the British prime minister to fight Britain's corner."
Conservative party MP Nadhim Zahawi, who was born in Baghdad, told BBC the ban meant he would be unable to visit his sons who are studying at Princeton University in the US.
London Mayor Sadiq Khan described Trump's move as "shameful", adding that "As a nation that, like the USA, values tolerance, diversity and freedom, we cannot just shrug our shoulders and say: 'It's not our problem'.