The memo, circulated privately to Republican senators on Wednesday and obtained by The New York Times, was also signed by Senator Bob Corker, Republican of Tennessee, another important figure in the debate. Mr. Corker voted for New Start when it was passed by the Senate Foreign Relations Committee in September, but now agrees with Mr. Kyl that it should not come to a floor vote during the current lame-duck session of Congress.

The White House argued that the problem predated Mr. Obama’s time in office. “We agree with Senators Kyl and Corker,” said Bob Jensen, a White House spokesman. “Modernization is needed. As the paper notes, the weapons complex was underfunded” for the five previous years. “It took several years of underfunding in the period before the president took office to get in this hole,” he said. “President Obama has a plan to get us out of it.”

Since Mr. Kyl’s statement last week, the White House has mounted a high-profile campaign to press the Senate to approve the treaty before the end of the year, making it a signal test of President Obama’s political strength at home after an election that cost his party control of the House as well as his credibility abroad as he tries to rebuild the relationship with Russia.

In an opinion article in The Wall Street Journal on Wednesday, Vice President Joseph R. Biden Jr. said that the stakes for the treaty were high. “Our uniformed military supports it,” he wrote. “Our European allies support it. Our national security interests are at stake. It is time for the Senate to approve New Start.”

The White House has been working with Mr. Kyl for months and contends that it has gone out of its way to address his concerns about modernization. It had already proposed spending $80 billion over 10 years on the nuclear complex and added $4.1 billion on Nov. 12 and a little more last week. White House officials felt blindsided by Mr. Kyl’s statement that not enough had been done to assuage him on the treaty.