The officials said the proposal would run through the late fall, and the price tag — approximately $18 billion — would be offset through cuts elsewhere in the budget so deficits would not rise.
Reid told reporters he was "cautiously optimistic" about a compromise emerging later in the day, and said he had held meetings with fellow Nevadan Dean Heller, a Republican, but provided no details.
But midafternoon, when Reid formally outlined the proposal, there was no evident Republican support for it, and each side accused the other of an unwillingness to compromise.
The expired law provided a maximum of 47 weeks of payments after an unemployed worker has exhausted state-funded benefits, usually 26 weeks.
The new measure would reduce the 47 weeks to a maximum of 31 weeks, officials said, based on a sliding scale that dates to the expired program.
Officials said the first tier of additional benefits would be six weeks, and be generally available to all who have used up their state eligibility.
They said an additional six weeks would be available in states where unemployment is 6 percent or higher; an additional nine weeks in states with joblessness of 7 percent or higher; and 10 more weeks in states where unemployment is 9 percent or more.
The cost would be offset in part by extending a previously-approved reduction in Medicare payments to providers, officials said, and in part by limiting or eliminating the ability of individuals on Social Security disability from also receiving unemployment benefits.
The officials who described the details of the possible legislation did so on condition of anonymity, saying they were not authorized to speak on the record.
Currently, Nevada and Rhode Island are the only states with unemployment of 9 percent.
Senators from the two states — Democrat Jack Reed of Rhode Island and Heller of Nevada — were central to the talks, and the White House was also being kept apprised.
Sen. Chuck Schumer, D-N.Y., told reporters that administration officials have indicated they would be satisfied with a deal that won the backing of Senate Democrats.
Any legislation that clears the Senate would also have to pass the House, where Speaker John Boehner, R-Ohio, has said he is only willing to consider an extension of the expired program that is fully paid for.