Under the compromise plan Russia would manufacture nuclear fuel for Iran and collect the waste to prevent the development of nuclear arms.
The plan has been welcomed by China and the chief of the UN International Atomic Energy Agency (IAEA) Mohamed ElBaradei.
Although the US State Department has expressed fears that Iran could be seeking to buy time through the Russian proposal, Mr Bush welcomed the plan.
"The Russians came up with the idea and I support it. And the reason why I think it makes sense is because I do believe people ought to be allowed to have civilian nuclear power,” Mr Bush said.
"However, I don't believe non-transparent regimes that threaten the security of the world should be allowed to gain the technologies necessary to make a weapon. And the Iranians have said, 'We want a weapon'. And it's not in the world's interests that they have a weapon,” he said.
Mr Bush stressed that the US wants to end the crisis through diplomatic efforts and the same message was given by US Secretary of State Condoleezza Rice in an address to the World Economic Forum in Davos, Switzerland.
She said Iran's nuclear program posed a "significant danger" and that the international community should "react strongly" and haul Tehran before the UN Security Council for action.
"Now when we're in the Security Council there will be many options available to the Security Council, but we believe that that is only the start of a new phase of diplomacy," she said.
The IAEA is to meet on February 2 in Vienna to discuss a Security Council referral.
The meeting was called by Britain, Germany and France (EU-3) after Iran this month resumed work on uranium enrichment, the process that makes fuel for civilian nuclear reactors, but can also be used to make atom bombs.
The US and the EU-3 are haggling with Russia and China over the wording of a draft resolution.
According to news agency AFP, the draft "recommends to the Security Council that it consider making clear to Iran that outstanding questions" can be resolved "by Iran responding" to calls by the IAEA for a suspension of nuclear fuel work and cooperation with IAEA inspectors.
Russia is believed to be resisting the clause because it fears Tehran's failure to meet the demand could open the way to sanctions.
Instead it favours a clause that requires the IAEA to just inform the Security Council about developments in Iran, without taking any action, a diplomatic source said.
'Covert operation'
The US and the European Union have expressed fears that Iran's nuclear program, which Tehran insists is for peaceful civilian purposes, could be a cover for developing nuclear weapons.
Washington has lobbied aggressively for international support to get the case referred to the UN Security Council.
China is reluctant to see UN action and on Thursday it backed the Russian plan as Iran's top security envoy met with Chinese officials in Beijing.
"We think the suggestion would be a good attempt at breaking the stalemate," Chinese foreign ministry spokesman Kong Quan said and reaffirmed China's opposition to sanctions.
Ali Larijani, secretary of Iran's Supreme National Security Council, said Wednesday in Moscow that the government "positively" welcomes the Russian plan which could be "perfected" in talks with Russian officials next month.
In Beijing, he held talks with Chinese Foreign Minister Li Zhaoxing and other senior officials on Thursday.
The European Union has also welcomed the Russian plan, while at the same time expressing scepticism about Iran's motives in accepting it.
EU external relations commissioner Benita Ferreo-Waldner called the Russian idea "interesting" but added: "Given its previous rejection, I think we are rather sceptical" about Iran's interest.