Trump said on Monday that Muslims nursed a "hatred" of America and should be banned from entering the country "until our country's representatives can figure out what is going on", following a deadly shooting in California.
The remarks caused dismay around the world, including among fellow Republican contenders for the presidency.
London Mayor Boris Johnson, a variety of other UK politicians and the Prime Minister's official spokesman have condemned the comments.
The Petitions Committee, made up of 11 cross-party backbench MPs, asks the relevant government department to give an official response once a petition receives 10,000 signatures. This is sent to all signatories to the petition.
After 100,000 signatures, the committee can ask for a debate to be held in Westminster Hall or it can request the Backbench Business Committee to hold a debate in the House of Commons.
Wherever the debate is held, it is expected that a relevant minister from the Government attend and respond to the MP leading the debate.
The committee can also "press the Government for action", ask a different Parliamentary committee to look into it, or invite the person who originated the petition to give evidence to Parliament.
So far this year, nearly 3,800 petitions have been submitted to the Government, of which 14 have been debated by Parliament and 61 have received an official government response. The 14 that were debated include whether to extend air strikes to Syria, changes to junior doctor contracts and banning the use of Neonicotinoids on crops.