Republican presidential candidate Ohio Gov. John Kasich said Tuesday he "profoundly disagrees" with the proposal.
"These are people, some of them are the worst of the worst. Why would we send them into our country?" Kasich told Fox News Tuesday. "I profoundly disagree."
Ben Carson, speaking to CNN's Poppy Harlow on Tuesday afternoon, said he wouldn't close Guantanamo until a better location was found to interrogate detainees.
"Where do we have that can take these prisoners of war?" Carson asked. "I'm not seeing what the alternative is, quite frankly."
Florida Sen. Marco Rubio conflated Obama's proposal to shut the facility with turning over the Guantanamo military base, where the prison is located, to the Cuban government.
"We are not giving back an important naval base to an anti-American, Communist dictatorship," Rubio said.
"We are not going to close Guantanamo," he said Tuesday while on the campaign trail in Las Vegas. "In fact, we shouldn't be releasing people there now. These are enemy combatants ... Not only are we not going to close Guantanamo, if we capture a terrorist alive, they are going to Guantanamo and we are going to find out everything they know."
Ben Rhodes, White House deputy national security adviser, said last week the administration is not planning to return Guantanamo to Cuba, saying it "is not on the table as a part of our discussions."
Texas Sen. Ted Cruz, also on the presidential campaign trail, also weighed in on the closing of Guantanamo, comparing the situation to when former President Jimmy Carter signed the Panama Canal Treaty, laying the groundwork for Panamanian control of the canal.
"Four decades ago Jimmy Carter gave away the Panama Canal, we built it, we paid for it and then a feckless left-wing president gave it away. Well Mr. President, you don't have the authority to give away vital military assets, and if you do that, you will be held accountable by the next president for undermining the national security of this country," said Cruz.
Capitol Hill opposition
The White House is already facing opposition from Republicans on Capitol Hill.
Rubio also announced he is co-sponsoring a bill aimed at preventing the transfer of the Naval Station at Guantanamo Bay to the Cuban government. Under the Naval Station Guantanamo Bay Protection Act, the President would have to receive congressional approval before "terminating, abandoning, or transferring the 45 square mile lease of land or waterways at the facility."
Further complicating matters for the White House, Obama recently signed the Defense Authorization Act and a Defense appropriations measure into law, which contain provisions preventing the transfer of Guantanamo detainees to the U.S. The White House is expected to say those provisions are unconstitutional.
Support from Sanders
Democratic presidential candidate Bernie Sanders, who visited the prison during a 2014 trip to Cuba, lauded Obama's proposal and sought to use the issue to contrast himself against rival Hillary Clinton.
"I am encouraged to see that the President is sending Congress a plan to shut down the Guantanamo Bay prison," he said in a statement. "As I have said for years, the prison at Guantanamo must be closed as quickly as possible. Others, including my opponent, have not always agreed with me."
Sanders' statement noted he was one of only three senators to vote in 2007 against barring the transfer of Guantanamo detainees to America; then-Sen. Clinton voted for the amendment that kept the prison open.