Mr. Kyl said he informed the Senate Democratic leader that there was not enough time to resolve all the issues during the lame-duck session that opened this week. “When majority leader Harry Reid asked me if I thought the treaty could be considered in the lame-duck session, I replied I did not think so given the combination of other work Congress must do and the complex and unresolved issues related to Start and modernization,” Mr. Kyl said in a written statement.

Mr. Kyl declined a request to be interviewed. Asked if the senator’s statement was meant to close the door to a lame-duck vote, his spokesman, Ryan Patmintra said: “Correct. Given the pending legislative business and outstanding issues on the treaty and modernization, there doesn’t appear to be enough time.”

Some Democrats hoped there was still an opening. “I talked with Senator Kyl today, and I do not believe the door is closed to considering New Start during the lame-duck session,” said Senator John Kerry of Massachusetts, the chairman of the Senate Foreign Relations Committee, who has presided over most of the 18 hearings on the treaty and secured endorsements from prominent Republican national security figures.

Mr. Kerry plans to hold a news conference on Wednesday with Secretary of State Hillary Rodham Clinton and Senator Richard G. Lugar of Indiana, a Republican treaty supporter. Mr. Biden said the administration would still support the additional $4.1 billion it offered Mr. Kyl on Friday.

Since a treaty requires a two-thirds vote, the White House needs at least eight Republicans in the departing Senate and thought it had a dozen ready to vote yes if Mr. Kyl assented. Without him, though, they began melting away. Senator Bob Corker of Tennessee, who voted for the treaty in committee, said Tuesday he now questions whether it’s “even practical for the administration to rush passage of the Start treaty during this lame-duck session.”

A Democratic leadership aide who spoke on the condition of anonymity to be more candid said, “If the Republicans’ lead negotiator says we shouldn’t consider Start during a lame duck, I think we have to take him at face value. Having said that, we are going to try and get it ratified in the lame duck.”

The Kremlin did not respond to the development, but Russian officials have expressed fear that Republican victories in this month’s midterm elections would damage relations. “We don’t have confidence that the document will secure enough votes,” Konstantin I. Kosachev, chairman of a parliamentary foreign affairs committee, said earlier in the day, according to the Russian news media. “The problem is not that the document is bad. We are confronting the fact that Republicans refuse to ratify the treaty.”