Mr Trump, who wants to bar all Syrian refugees from entering the United States, made the pledge on the eve of a crucial vote in the race for the Republican presidential nomination.
The state of New Hampshire will on Tuesday become the second American state to make its decision and the billionaire leads by a large margin in polls.
At an usually small campaign event in the town of Salem about 200 people packed into a modest brick building and were allowed to ask the candidate questions.
One man said he lived in Greenwich, Connecticut, where Mr. Trump has a home, and there were plans to relocate Syrian refugee families there.
He asked Mr. Trump if he could “look children aged five, eight, ten, in the face and tell them they can’t go to school here”.
Mr. Trump did not hesitate and said he could, which brought applause from the crowd.
He said: “I can look in their faces and say ‘You can’t come’. I’ll look them in the face.”
They may be Isil, they may be Isil-related. It could be a Trojan horse. If two per cent of those people are bad the trouble is unbelievable.”