The same proposal would reverse a provision approved in December to slow the annual increase in veterans retirement benefits for recipients under age 62.
Reid, in comments to reporters, said unemployment benefits had been extended several times when George W. Bush was president and Congress did not insist on paying for them with cuts elsewhere in the budget.
"This is new religion to them," he said of Republicans.
Any debate over paying for renewing jobless benefits is almost certain to circle back to a perennial disagreement over taxes.
In last month's successful negotiations over spending legislation, Democrats sought to close tax loopholes to keep deficits from rising. Republicans refused, demanding spending cuts or higher fees instead.
At the same time the two parties struggle with one another, Republicans are also under pressure from outside groups who oppose any renewal of jobless benefits, including some with ties to the tea party.
Any legislation that clears the Senate would also have to make it through the House, where dozens of tea party-aligned lawmakers are in office.
In a statement issued shortly after the Senate vote, Boehner, R-Ohio, said he has previously informed the White House that any measure to renew unemployment benefits "should not only be paid for but include something to help put people back to work. To date, the president has offered no such plan. If he does, I'll be happy to discuss it."
At the White House, Obama, too, said he was siding with victims of the recession.
"These aren't folks who are just sitting back, waiting for things to happen," he said. "They're out there actively looking for work."
At issue is a system that provides as much as 47 weeks of federally funded benefits, beginning after the exhaustion of state benefits, usually 26 weeks in duration.
The first tier of additional benefits is 14 weeks and generally available to all who have used up their state benefits.
An additional 14 weeks is available in states where unemployment is 6 percent or higher. Nine more weeks of benefits are available in states with joblessness of 7 percent or higher. In states where unemployment is 9 percent or higher, another 10 weeks of benefits are available.
Officials also said a little-noticed provision in the legislation is specifically designed to benefit the long-term unemployed in North Carolina, where Democratic Sen. Kay Hagan faces a stiff challenge for a new term.
It would make residents eligible for long-term benefits by permitting the state to negotiate an agreement with the Department of Labor. State residents are currently ineligible because of state benefits were reduced below a federal standard.
Associated Press writer Donna Cassata contributed to this story.