In response, Boehner's spokesman said Reid should stop talking and instead take up legislation passed by the House that would avert the fiscal cliff.
"The House has already passed legislation to avoid the entire fiscal cliff. Senate Democrats have not," said the spokesman, Michael Steel.
Reid and Democrats reject the GOP proposals, which would extend all the Bush tax cuts and revamp the spending cuts of the fiscal cliff, as insufficient steps that would shift too much of the burden of deficit reduction on the middle class.
Instead, Reid called on Boehner to allow a vote on a Senate-passed measure that would implement Obama's plan to extend tax cuts to the $250,000 threshold. However, McConnell rejected that possibility on Thursday in seeking to focus the debate on revising House-passed measures.
Rep. Nan Hayworth, R-New York, acknowledged Wednesday that a deal will have to include some form of higher rates on top income brackets, but she said her party would fight to make it as minimal as possible.
Hayworth also made clear to CNN that a limited agreement was the most to expect for now, saying: "I don't think we're going to get the big plan in the next six days."
A statement Wednesday by Boehner's leadership team said the Senate must act first on proposals already passed by the House but rejected by Senate leaders and Obama.
"If the Senate will not approve and send them to the president to be signed into law in their current form, they must be amended and returned to the House," the leadership statement said. "Once this has occurred, the House will then consider whether to accept the bills as amended, or to send them back to the Senate with additional amendments. The House will take this action on whatever the Senate can pass, but the Senate first must act."
Obama and Democrats have leverage, based on the president's re-election last month and Democratic gains in the House and Senate in the new Congress. In addition, polls consistently show majority support for Obama's position on taxes.
The Gallup daily tracking poll released Wednesday showed 54% of respondents support Obama's handling of the fiscal cliff negotiations, compared with 26% who approve of Boehner's performance.
A senior Senate Democratic source told CNN on Wednesday that Reid has made clear in private conversations that he will need assurance that any plan can pass both the Senate and the House before he will bring it up.
"It is to nobody's advantage to have a failed Senate vote at this point," the source said on condition of not being identified. "This will be the last train we will have, and there is no sense in it leaving the station before we have assurance it will get through."
Remaining questions include whether enough Republicans will support a compromise acceptable to Democrats, and whether McConnell and Senate Republicans will allow a simple majority vote to take up and pass any proposal, or stick to the filibuster level of 60%.
"We believe very strongly a reasonable package can get majorities in both houses," a senior White House official said. "The only thing that would prevent it is if Senator McConnell and Speaker Boehner don't cooperate."
Some Senate Democrats have discussed holding off on bringing up a proposal until the final days of 2012 to increase pressure on Republicans to support avoiding higher taxes on everyone due to the fiscal cliff.
While the focus now is on a possible agreement in coming days or weeks, anti-tax crusader Grover Norquist told CNN this week that the nation should gird for long-range battle.
"It's four years of a fight. It's not one week of a fight," said Norquist, who has threatened to mount primary challenges against Republicans who violate a pledge they signed at his behest against ever voting for a tax increase.
He predicted "a regular fight" when Congress needs to authorize more government spending and raise the federal debt ceiling in coming months.
"There the Republicans have a lot of clout because they can say we'll let you run the government for the next month, but you've got to make these reforms," he said.
On Wednesday, Treasury Secretary Timothy Geithner informed Congress that the government would reach its borrowing limit at the end of the year -- in five days' time -- but could take steps to create what he called "headroom" for two months or so.
However, Geithner said uncertainty about the fiscal cliff negotiations and possible changes to the deficit situation made it difficult to predict precisely how long the government's steps to ease the situation would last.
The GOP opposition to any kind of tax rate increase has stalled deficit negotiations for two years and led to unusual political drama, such as McConnell recently filibustering a proposal he introduced and last week's rebuff by House Republicans of the alternative tax plan pushed by Boehner, their leader.
Reid and other Senate Democrats say House Republicans must accept that an agreement will require support from legislators in both parties, rather than a GOP majority in the House pushing through a measure on its own.
Some House Republicans have said they would join Democrats in supporting the president's proposal in hopes of moving past the volatile issue to focus on the spending cuts and entitlement reforms they seek.
The possibility of a fiscal cliff was set in motion over the past two years as a way to force action on mounting government debt.