The Pentagon submitted a plan to Congress on Tuesday laying out how the Obama administration would close the Guantanamo Bay prison in Cuba.

The proposal met a deadline set by lawmakers for the White House to explain how it would close the controversial prison camp, but pushback against some of the details in the proposal was already building on Capitol Hill, prompting the White House press secretary to urge Congress to keep what he called, an "open mind."

CBS News correspondent Margaret Brennan says it is likely to be President Obama's last attempt to get Congress to shut down Guantanamo -- an action he vowed to take during his first campaign for the presidency more than seven years ago.

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Mr. Obama has argued that keeping Guantanamo open undercuts U.S. national security, as the indefinite detention of prisoners without trial and the harsh interrogation techniques used there turned the facility into a recruiting tool for terrorists.

However, the Obama administration seems to have underestimated how politically fraught it would be to get the facility shuttered.

Right now, 91 prisoners remain at Guantanamo.

The administration plans to transfer 35 of them -- mostly Yemeni nationals -- to another country in the next year.

The remaining 56 -- including Khalid Sheikh Mohammed, the mastermind of the Sept. 11, 2001 terror attacks on New York and Washington -- are either considered too great a security threat to transfer out of U.S. custody, or would face trial in criminal or military courts.

All of them would still be held by military guards at one of three U.S. sites under consideration; the Charleston, South Carolina Naval brig, the Fort Leavenworth, Kansas military prison, and a supermax prison near Fort Carson, Colorado.

‎The White House will argue that those options will save taxpayers money, since they are projected to cost less than the $400 million it costs each year to keep Guantanamo open.

But as Brennan reports, there's a problem: Under current U.S. law, it is illegal to bring these prisoners to the U.S., so unless the Republican-controlled Congress changes that law, the proposal is dead on arrival.

That leaves President Obama with a choice: Either take the politically-risky move of shutting down Guantanamo through executive action, or admit defeat.

In an interview Tuesday on WMAL Radio, the chair of the House Armed Services Committee, Texas Republican Mac Thornberry, said he was at least willing to hold a hearing on the president's plan.

"I'm not saying it's impossible (to close Guantanamo)," Thornberry said, but adding: "There has to be a specific plan, and there has to be a lot of specific elements in that plan."

"How you're going to do this, where (detainees are) going to go, what it's going to cost. What do you do if you capture a (ISIS leader Abu Bakr) Bagdhadi tomorrow? Where's he going to go, and how are you going to prosecute him?" the Representative asked in the radio interview.

Thornberry said he feared the Pentagon proposal "may be more of a P.R. exercise than a real plan to work with Congress to do this."