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President Obama and House Speaker John Boehner spoke by phone early Tuesday evening, senior administration officials told Fox News, after both sides exchanged new proposals on how to avoid the looming year-end fiscal crisis over tax rates and spending cuts.

Obama and Boehner had met Sunday at the White House. The White House then offered a new proposal lowering its proposed figure for tax revenue from $1.6 trillion to $1.4 trillion, sources told Fox Business. Boehner offered a Republican counter proposal on Tuesday, his office said, but it didn't provide details.

While Tuesday's phone conversation and the exchange of offers indicated some movement on the path to an agreement, it was unclear whether any real progress was being made.

Rep. Sandy Levin of Michigan, the top Democrat on the House Ways and Means Committee, suggested that $1.4 trillion was a revenue level Democrats could work with, telling Fox News, "it's basically in the ballpark."

However, Democrats want to see Boehner's offer and are very suspicious of what he said on the floor this morning.

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Boehner said the meeting at the White House Sunday afternoon was "cordial" -- but Republicans are "still waiting for the White House to identify what spending cuts the president is willing to make as part of the balanced approach that he promised the American people."

Boehner said he remains "hopeful" a deal can be reached, but said Obama has an "obligation" to put forward a new and specific plan if he objects to the plan Republicans proffered last week.

"The longer the White House slow-walks this process, the closer our economy gets to the fiscal cliff," he said. "We know that the president wants more stimulus spending and an increase in the debt limit without any cuts or reforms. That's not fixing our problem -- frankly, it's making it worse."

The White House, though, rejected Boehner's claims.

"The president, unlike any other party to these negotiations, has put forward detailed spending cuts as well as detailed revenue proposals," White House Press Secretary Jay Carney said Tuesday.

Senate Democratic Leader Harry Reid, for his part, splashed cold water on the prospects of reaching a budget deal by Christmas to avert sweeping tax hikes and spending cuts, saying "it's going to be extremely difficult" to do so by Dec. 25.

Boehner did not indicate whether he and his team are considering any changes to the Republican negotiating position. Boehner, while putting revenue on the table, has all along rejected Obama's demand that tax rates increase next year on the top 2 percent.

A few Republicans in both chambers, however, have urged Boehner to give the president what he wants on tax rates -- and focus energy on fighting for entitlement cuts. Without discussing taxes in great detail, Boehner and Senate Republican Leader Mitch McConnell both pressed Obama Tuesday for more details on possible cuts.

"The president seems to think that if all he talks about are taxes, and that's all reporters write about, somehow the rest of us will magically forget that government spending is completely out of control, and that he himself has been insisting on balance," McConnell said in remarks on the Senate floor.

The automatic tax increases and spending cuts will kick in after the first of the year, and economists have warned that they could plunge the nation into another recession.

The two sides, though, have struggled to reach a deal. Each has accused the other of being too vague with its plan. Just as Republicans are now pressing Obama for details on spending cuts, Obama has criticized Republicans for withholding details about which deductions they'd be willing to trim.

Fox News' Chad Pergram, Fox Business' Rich Edson and the Associated Press contributed to this report.