U.S. officials in Washington have only said the final language is close.
White House Spokesman Robert Gibbs said an agreement would not be finalized until President Barack Obama and Russian President Dmitry Medvedev have a chance to speak again - likely in the next few days, reports CBS News White House correspondent Mark Knoller.
Mr. Obama spent an hour Wednesday in the White House with Democratic Sen. John Kerry, chairman of the Senate Foreign Relations Committee, and Sen. Richard Lugar, the committee's ranking Republican. Both would play major roles in Senate ratification of the emerging treaty.
Czech officials announced earlier Wednesday that Prague will host the signing of the new U.S.-Russian treaty to reduce long-range nuclear weapons that would replace the 1991 Strategic Arms Reduction Treaty.
The Kremlin source, speaking by telephone to The Associated Press, said the documents included the treaty and protocol. Russian Foreign Minister Sergey Lavrov said last week that the treaty was 20 pages long, with an extensive protocol attached.
The new treaty would most likely be signed in Prague, but the time and place would be set by the two presidents, the Kremlin official said. The official said there would be no further information from the Kremlin until Obama and Medvedev had spoken by telephone in the coming days.
The Russian ambassador to Prague, Alexey Fedotov, told Czech President Vaclav Klaus the date for the signing, which will be released later, Klaus' office said in a statement.
Prague was the city whereto seeking "a world without nuclear weapons."
As part of that strategy, Mr. Obama shook hands with Medvedev last summer on plans to reduce sharply both countries' nuclear stockpiles. The two leaders had hoped to enshrine the new limits in a replacement for the 1991 START accord, but that treaty expired in December as the talks dragged on.
Negotiations, which have been under way in Geneva, are centered on disputes over verification measures and Russia's objection to U.S. missile defense plans for Europe.
A senior U.S. administration official in Washington, speaking on condition of anonymity because no deal has been announced, confirmed reports that Prague would host the signing. The official added that the meeting with Kerry and Lugar was "part of our ongoing consultations with Congress on START negotiations."
Russian negotiators have balked at including some intrusive weapons verification measures in the new treaty. The Obama administration has warned that without these, Senate ratification could prove difficult.
Any agreement would need to be ratified by the legislatures of both countries and would still leave each with a large number of nuclear weapons, both deployed and stockpiled.
The expired START treaty, signed by Soviet President Mikhail Gorbachev and U.S. President George H.W. Bush, required each country to cut its nuclear warheads by at least one-fourth, to about 6,000, and to implement procedures for verifying that each side was sticking to the agreement.
The two sides pledged to continue to respect the expired treaty's limits on nuclear arms and allow inspectors to continue verifying that both sides were living up to the deal.
Mr. Obama and Medvedev agreed in July to cut the number of nuclear warheads each possesses to between 1,500 and 1,675 within seven years as part of a broad new treaty.