The Michigan group, Leaders Advancing and Helping Communities, in Dearborn, said it would also reject $500,000 in federal aid. Suehaila Amen, an executive board member, said that her group had already been discussing before the Reuters report came out whether to accept the grant, because some in the community see the hotly disputed program as a way to spy on Muslims. But the decision was sealed by Mr. Trump’s statements and actions, Ms. Amen said.

“You’re talking about a president who has maligned the Muslims throughout the entire campaign,” she said. “We don’t feel like we need to compromise our integrity, transparency and integrity in the community for some money or funding.”

Another organization, the Muslim Public Affairs Council Foundation, which has offices in Washington and Los Angeles, said it was waiting to see whether the administration moved forward with changes to the program before deciding whether to keep its nearly $400,000 grant.

And Jihad Turk, the president of Bayan Claremont, a graduate college in Claremont, Calif., for Muslim scholars and religious leaders, said it would not accept its $800,000 grant, one of the largest awarded, if the administration renamed the program and entirely shifted its focus.

The Department of Homeland Security declined to comment, though people briefed on the matter said there was fierce resistance inside the department to making changes to the program that could further alienate Muslim communities assisting antiterrorism efforts. During his confirmation hearing as department secretary last month, John F. Kelly stressed the need to build trust with Muslim communities.