And it calls for an investigation into transferring the facility's remaining prisoners, currently numbering about 245, to similar facilities in the US.
Anyone covered by the order still in detention when the prison is closed "shall be returned to their home country, released, transferred to a third country, or transferred to another US detention facility", Reuters quoted the draft as saying.
A senior US official said Obama was unlikely to sign and issue the order on his first day in office but he is expected to sign it soon.
|Khalid Sheikh Mohammed faces war crimes
charges over the September 11 attacks [EPA]
The reports come after a US judge at the Guantanamo Bay prison camp stopped the military trial of five men accused of planning the September 11 attacks on the US following a request by the new US president, who asked for all 21 Guantanamo trials be suspended for 120 days.
The halt came despite pleas from four of the five 9/11 suspects, who face war crimes charges over the attacks, for the trials go ahead and for them to be permitted to plead guilty.
The defendants include Khalid Sheikh Mohammed, who confessed to planning the attacks after he was subjected to harsh interrogation techniques known at Guantanamo including "waterboarding", which simulates the sensation of drowning.
Mohammed, Ramzi Binalshibh and three others, Mustafa Ahmed al-Hawsawi, Walid bin Attash and Mohammed's nephew, Ali Abdul Aziz Ali, face 2,973 counts of murder, one for each person killed when hijackers crashed passenger jets into the World Trade Center, the Pentagon and a Pennsylvania field on September 11, 2001.
Earlier on Wednesday, the trial of Omar Khadr, a Canadian detainee, was also suspended for 120 days.
Khadr, who was arrested in Afghanistan at the age of 15, is accused of killing a US soldier with a grenade in Afghanistan in 2002 and has been held for seven years.
"The practical effect of today's ruling is to pronounce the military commissions process dead"
Omar Khadr's defence lawyer
Navy Lieutenant Commander William Kuebler, Khadr's defence lawyer, said: "The practical effect of today's ruling is to pronounce the military commissions process dead."
Other judges trying Guantanamo cases are expected to rule on the presidential call.
Obama's request was among his first actions as president, after being inaugurated on Tuesday.
Jamil Dakwar, the director of the human rights programme at the American Civil Liberties Union, said Obama's request was a positive step but "the president's order leaves open the option of this discredited system remaining in existence".