"There is too much at stake," said Daisy Khan, who, with her husband, Imam Feisal Abdul Rauf, is leading the project. "Constitutional rights, the development of the Muslims here, how the world is watching the United States. We tell people America upholds religious freedom. We should not compromise those values."
Khan spoke to Sally Quinn of the Washington Post, who quoted her in a blog post.
"We have to convince people that not all Muslims are extremists," said Khan, as quoted by Quinn. "We have to educate them on being able to distinguish between us and on the issue of Islamophobia. This is a bigger fight. This is a defining moment for us."
The project, which is called Park51, has created a national debate over religious tolerance and sensitivity to the victims of the 9/11 attacks. The proposed center is two blocks from the site of the World Trade Center. It has often been referred to as the "Ground Zero Mosque," although its supporters say it would mainly be a community center, not principally a mosque.
Critics charge that having what they call a mosque so close to what they consider hallowed ground is an insult to the victims' families, especially because the attack was perpetrated in the name of Islam.
"It's hard for us to imagine we are in the thick of a controversy like this," Khan told Quinn. "The Republicans are really going after us."
Rauf, her husband, is on a 15-day Mideast tour funded by the U.S. State Department, where he said today he hopes to draw attention to the common challenges to battle radical religious beliefs.
Speaking in Manama, Bahrain, Rauf refused to talk about the New York project, but warned about what he said was the larger problem of extremism and religious intolerance.
"This issue of extremism is something that has been a national security issue, not only for the United States but also for many countries and nations in the Muslim world," he said.
The imam is a Sufi Muslim, a mystical branch of Islam whose adherents have been attacked by Muslim extremists overseas.
Oz Sultan, a spokesman for the planned center in New York, said the groups behind Park51 had recently been in touch with New York Gov. David Paterson's office to discuss the governor's reported proposal to move the center to a parcel of land owned by the state. Earlier in the week, the groups said they were unaware of any discussions with Paterson.
"There's been an initial contact and I know a conversation is ongoing," Sultan said, declining to discuss the details.
Paterson's office did not return calls for comment from ABCNews.com, but he told CNN's "Larry King Live" Wednesday that he is open to finding a compromise site with the Islamic center's developers.
"If people put their heads together, maybe we could find a site that's away from the site now but still serves the ... area," he said. "That would be a noble gesture to those who live in the area who suffered after the attack on this country, and at the same time would probably in many ways change a lot of people's minds about Islam, which is really a peaceful religion practiced by peace-loving people.
"There's no attempt at pressure or coercion here. I'd just like to talk about what might be a magic moment in our history."
Meanwhile, House Speaker Nancy Pelosi, D-Calif., has called for an investigation into funding, not for the center, but for the conservative effort to oppose it.
"There is no question that there's a concerted effort to make this a political issue by some," Pelosi told San Francisco radio station KCBS. "I join those who have called for looking into how is this opposition to the mosque being funded."
The Associated Press contributed to this story.