Texas Sen. John Cornyn says Obama is "disconnected from mainstream America" and that voters this fall will "render their verdict." Cornyn leads the GOP's Senate campaign committee.
Obama has said that religious freedom allows the mosque to be built. But he says he's not commenting on "the wisdom" of building a mosque two blocks from ground zero.
Democratic Sen. Jack Reed of Rhode Island says the economy will remain the dominant election issue.
Saturday, President Obama said Muslims have the right to build a mosque near Ground Zero, but he did not say whether he thought it was a good idea.
Obama commented during a trip to Florida, where he expanded on a Friday night White House speech asserting that Muslims have the same right to freedom of religion as everyone else in America.
The president's statements thrust him squarely into a debate that he had skirted for weeks and could put Democrats on the spot three months before midterm elections where they already were nervous about holding control of the House and maybe even the Senate. Until Friday, the White House had asserted that it did not want to get involved in local decision-making.
New York City Mayor Michael Bloomberg, an independent who has been a strong supporter of the mosque, welcomed Obama's White House speech as a "clarion defense of the freedom of religion."
Gov. Charlie Crist, R-Fla., who was among those who met with Obama on Saturday, lauded the president's position.
"I think he's right - I mean you know we're a country that in my view stands for freedom of religion and respect for others," Christ said after the Florida meeting with Obama and other officials. "I know there are sensitivities and I understand them.
This is a place where you're supposed to be able to practice your religion without the government telling you you can't."
Others were quick to pounce.
Democratic U.S. Senate candidate Jeff Greene of Florida took Obama's Friday speech to mean the president supports the construction.
"President Obama has this all wrong and I strongly oppose his support for building a mosque near Ground Zero especially since Islamic terrorists have bragged and celebrated destroying the Twin Towers and killing nearly 3,000 Americans," Greene said. "Freedom of religion might provide the right to build the mosque in the shadow of Ground Zero, but common sense and respect for those who lost their lives and loved ones gives sensible reason to build the mosque someplace else."
The mosque would be part of a $100 million Islamic community center two blocks from where nearly 3,000 people perished when hijacked jetliners slammed into the World Trade Center towers on Sept. 11, 2001.
The proposed construction has sparked debate around the country that included opposition from top Republicans including Sarah Palin and Newt Gingrich as well as the Jewish civil rights group the Anti-Defamation League.
Obama's Friday comment was taken by some to mean that he strongly supports the building of an Islamic center near the site of the Sept. 11 terrorist attack, something he never actually said.
Speaking to a gathering at the White House Friday evening to observe the Islamic holy month of Ramadan, Obama said that he believes "Muslims have the same right to practice their religion as everyone else in this country."
"That includes the right to build a place of worship and a community center on private property in lower Manhattan, in accordance with local laws and ordinances," he said. "This is America, and our commitment to religious freedom must be unshakable."
Asked Saturday about the issue during his trip to Florida, Obama said: "I was not commenting and I will not comment on the wisdom of making a decision to put a mosque there. I was commenting very specifically on the right that people have that dates back to our founding."
Obama said that "my intention was simply to let people know what I thought. Which was that in this country we treat everybody equally and in accordance with the law, regardless of race, regardless of religion."
Some relatives of people killed in the Sept. 11 attacks supported Obama's comments.
The mosque is "in many ways ... a fitting tribute," said Colleen Kelly of the Bronx, who lost her brother Bill Kelly Jr. in the attacks.
"This is the voice of Islam that I believe needs a wider audience," said Kelly, who is Catholic. "This is what moderate Islam is all about."
Opinions are mixed among family members.
Sally Regenhard, whose firefighter son was killed at the World Trade Center, has said the president's comments show "a gross lack of sensitivity to the 9/11 families and to the people who were lost."