The Pentagon proposal suggests 13 potential sites on US soil for the transfer of remaining detainees but does not identify the facilities or endorse a specific one, administration officials said.
Mr Obama pledged to close the prison and move the detainees as a candidate for the White House in 2008.
Lawmakers largely oppose moving the prisoners to the US, however, and Mr Obama’s final attempt to get congressional backing is unlikely to gain traction.
“Let us go ahead and close this chapter,” Mr Obama said in White House remarks. “I don’t want to pass this problem on to the next president, whoever it is.”
“Keeping this facility open is contrary to our values. It undermines our standing in the world. It is viewed as a stain on our broader record of upholding the highest standards of rule of law.”
Mr Obama leaves office in January 2017.
The Guantanamo prisoners, held at a US naval station in south-eastern Cuba, were detained by US troops in Iraq and Afghanistan.
The facility came to symbolise aggressive detention practices in years past that opened the US to allegations of torture.
The transfer and closure costs would be $290m to $475m (about €430m), an administration official said.
Housing remaining detainees in the US would be $65m to $85m cheaper than at the Cuba facility, the official said, so costs would be offset in three to five years.
Some 35 prisoners will be transferred from Guantanamo to other countries this year, leaving the final number of prisoners below 60, officials said.
Mr Obama is considering closing the facility by executive order if lawmakers do not back his proposal.
The plan would send detainees who have been cleared for transfer to their homelands or third countries and transfer remaining prisoners to US soil to be held in maximum-security prisons.
Congress has banned such transfers to the US since 2011.
Though the Pentagon has previously noted some of the sites it surveyed for use as potential US facilities, the administration wants to avoid fuelling any political outcry in important swing states before the November 8 presidential election.
Late last year, assessments done by the Pentagon team suggested that the Centennial Correctional Facility in Colorado would be a more suitable site to send detainees whom officials believe should never be released.
Those officials were not authorised to discuss that matter publicly, so spoke on condition of anonymity.