WASHINGTON — A top Senate Democrat proposed Tuesday to give unemployed people jobless benefits through Memorial Day instead of risking another cutoff in just three weeks.

Max Baucus, D-Mont., proposed the additional time as the Senate officially began debate on legislation to revive a federal unemployment insurance program for people who have been out of a job for more than six months.

Ohio Republican George Voinovich emerged as a key figure as Democrats and Republicans continued to quarrel over whether federal jobless benefits for the long-term unemployed should be paid for with borrowed money. Voinovich signaled he will side with Democrats to provide a crucial vote that would ensure the measure’s speedy advance into law — even though it would add about $18 billion to the deficit.

Extending the benefits has prompted a battle between Republicans promising to battle the deficit and Democrats insisting that the government continue to borrow the money to pay for checks averaging $335 a week.

The GOP move is unprecedented; more than half of Senate Republicans voted just last month for an earlier debt-finance extension of jobless benefits. But the issue of deficits and debt is of increasing concern to voters who in only seven months will determine if Republicans take back control of Congress.

Four Republican senators helped Democrats defeat a GOP filibuster on Monday: Susan Collins and Olympia Snowe of Maine, Voinovich, and Scott Brown of Massachusetts. But their votes are not guaranteed on subsequent procedural hurdles, including a vote to waive budget rules and add the cost of the jobless benefits to the $12.8 trillion national debt.

Democrats need to pick up at least one Republican to advance the measure. Last month, in a similar situation, Collins was that single Republican as she helped Democratic leaders maneuver their way through a tight spot. Now she says she will insist that the temporary unemployment benefits bill be “paid for” so as to not increase the deficit.

But it’s sounding as if Voinovich may step in to help Democrats.

“All I know is that I’ve got two guys on my street that are unemployed,” Voinovich said in an interview. “This unemployment (compensation) is a big deal. I hate borrowing the money for it. But . . . it’s allowed people to keep their families together.”

GOP leaders seek to derail the measure because it runs astray of budget rules. A key tally to waive those rules — requiring 60 votes in the 100-member Senate — is slated for midday today.

Unemployment compensation has ended for more than 400,000 people whose benefits lapsed but who would have otherwise been eligible to reapply for additional weeks of compensation if the program’s authority had not ended last week.