Mr Gates will retain his post when the Obama administration takes office.
The Guantanamo facility currently holds about 250 inmates without charge, on suspicion of being involved in terrorism.
Pentagon spokesman Geoff Morrell said Mr Gates had instructed his staff to investigate the best way to go about shutting down the prison.
'If this is one of the president-elect's first orders of business, the secretary wants to be prepared to help him as soon as possible,' Mr Morrell told reporters.
'(Mr Gates) has asked his team for a proposal on how to shut it down - what would be required specifically to close it, and move the detainees from that facility, while at the same time, of course, ensuring that we protect the American people from some dangerous characters.'
President-elect Obama has pledged to close the prison located at the US naval station in southeastern Cuba, which has come to symbolise aggressive detention practices that opened the US to allegations of torture.
There was no immediate word on how soon Mr Obama might address the Guantanamo question. But the President-elect, who has repeatedly called its closure a top foreign policy priority, said in the current issue of Time magazine that he hoped to have the jail shut during the first two years of his term.
11 September accused among inmates
Mr Gates, who was appointed by George W Bush but has agreed to stay on under Barack Obama, also wants Guantanamo shut.
The detainees apprehended as part of Mr Bush's 'war on terror' include Khalid Sheikh Mohammed, accused of masterminding the 11 September attacks.
Defence officials said the task of closing Guantanamo is likely to be a lengthy, complex process that would involve all three branches of the government.
'You look at this sort of thing early because that's when you have momentum for bringing about change. With a new administration coming in, you've got people who are willing to do what's necessary,' said one defence official.
Officials said members of the Office of the Secretary of Defence and the Joint Chiefs of Staff will provide President-elect Obama with a set of options for tackling the complex issues raised by Guantanamo.
It is not clear whether the Pentagon will recommend a specific course of action to the new president.
The Obama administration would need to decide where to hold current detainees, particularly about 110 prisoners who the Pentagon believes are too dangerous to be released from US custody.
Options might include military installations on US soil and civilian federal prisons.
The US would also need to decide what kind of court system should handle trials for roughly 80 detainees. There are now charges against 20.
'The request has been made, his team is working on it so that he can be prepared to assist the president-elect should he wish to address this very early in his tenure,' the press secretary said.
Addressing these issues could require input from a number of other government entities including the Justice Department, judicial officers and Congress, officials said.
In fact, Mr Gates has said that Congress should pass legislation to protect the US public by preventing any former Guantanamo detainee from living in the US.
The Guantanamo tribunals are scheduled to reconvene on 19 January for pre-trial hearings for Canadian captive Omar Khadr, who is set for trial the following week on charges of murdering a US soldier with a grenade during a firefight in Afghanistan.
