The issue was raised as a point of order by Labour MPs Yvette Cooper and Stephen Doughty, who asked the Home Secretary, Amber Rudd, to respond to the tweets.
Mr Doughty said: "It appears that the President of the United States has, in recent moments, been retweeting comments from far-fight organisation Britain First - highly inflammatory videos, including some posted by an individual who I believe has recently been arrested and charged relating to certain serious offences."
He asked the Speaker, John Bercow, if there were plans for Ms Rudd or the Foreign Secretary, Boris Johnson, to make a statement on the issue.
Ms Cooper, a former Shadow Home Secretary, said the "significance and the seriousness" of Mr Trump giving Britain First "a huge platform" meant it was "appropriate" for government ministers to condemn the Republican.
Mr Bercow gave Ms Rudd the opportunity to respond but she declined to do so, prompting MPs to shout: "Go on, stand up."
Others yelled "absolutely disgraceful" when the issue of Mr Trump's tweets was mentioned.
The Speaker said it would be "wrong to expect a government minister to respond immediately".
It comes after Mr Trump retweeted three videos posted by Jayda Fransen, Britain First's deputy leader.
The first video claimed to show "Muslim migrants beating up a Dutch boy on crutches", while the second was captioned "Muslim destroys statue of Virgin Mary". The third read "Islamist mob pushes teenage boy off roof and beats him to death".
The posts sparked widespread condemnation, with critics accusing the US President of giving a platform to Britain First and Fransen, who has previously been convicted of religiously aggravated harassment over her treatment of a Muslim woman wearing a hijab. She was arrested again earlier this month over a speech she gave a rally in Belfast.
Other MPs used social media to add their condemnation of Mr Trump's tweets. Chris Bryant wrote: "Donald Trump is inciting religious hatred in this country by retweeting the convicted racist."
Tottenham MP David Lammy said: "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. Donald Trump you are not welcome in my country and my city."
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies