A Twitter account associated with the mysterious hacking group wrote: "Trump Towers NY site taken down as statement against racism and hatred."
The account, which has more than 72,000 followers, added: "Shame (not) your site is down, Donald Trump."
In a video posted on YouTube, a member of the collective said: "Donald Trump think twice before you speak anything. You have been warned Mr. Donald Trump
They added: "The more the United States appears to be targeting Muslims, not just radical Muslims, you can be sure that ISIS will be putting that on their social media campaign."
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| Trump has called for Muslims to be banned from entering the US |
They mocked the death cult, also known as Daesh, by Photoshopping rubber ducks and pigs into ISIS propaganda images.
They also disabled social media accounts belonging to suspected ISIS militants – who they have branded "Daeshbags".
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| Anonymous recently waged war on Islamic State |
An Anonymous member, known only as TMT, spoke exclusively to new4news911 before the day of trolling.
She said: "We hold down their communications quite well.
"We not only have their accounts suspended but we also are able to disable their websites so they are not able to do as much recruiting that way.
"They can't spread their propaganda as well. It slows them down."
The Prime Minister slammed his "divisive" views while Mr Johnson, the London Mayor, branded them "utter nonsense".
And Mrs Clinton, his biggest rival to replace Barack Obama as president next year, said he was "shameful and wrong".
A spokesman for Trump Towers was not available for comment