WASHINGTON (AP) — In a test of divided government, President Barack Obama and House Speaker John Boehner sought an elusive compromise Tuesday to prevent economy-damaging year-end tax increases for the middle class, speaking by phone after a secretive exchange of proposals.

Details were sparse and evidence of significant progress scarcer still, although officials said the president had offered to reduce his initial demand for $1.6 trillion in higher tax revenue over a decade to $1.4 trillion.

There was no indication he was relenting on his insistence — strongly opposed by most Republicans — that tax rates rise at upper incomes.

Boehner sounded unimpressed in remarks on the House floor at midday, well before he and the president talked by phone about attempts to avert a ‘‘fiscal cliff,’’ across-the-board tax increases and cuts in defense and domestic programs that threaten to send the economy into recession.

‘‘The longer the White House slow-walks this process, the closer our economy gets to the fiscal cliff,’’ he said, declaring that Obama had yet to identify specific cuts to government benefit programs that the president would support as part of an agreement that also would raise federal tax revenue.

In rebuttal, the White House swiftly detailed numerous proposals Obama has made to cut spending, including recommendations to cull $340 million from Medicare over a decade and an additional $250 billion from other government benefit programs.

The House Democratic leader, Rep. Nancy Pelosi of California, challenged Boehner to allow a vote on the president’s proposal to extend most expiring tax cuts while letting them lapse at higher incomes.

She predicted it would gain ‘‘overwhelming approval,’’ even in the GOP-controlled House.

Two weeks before the year-end holidays, time to find agreement was short, but not prohibitively so.

‘‘I think it’s going to be extremely difficult to get it done before Christmas but it could be done,’’ said Senate Majority Leader Harry Reid.

Boehner’s office took the step — unusual in secretive talks — of announcing that Republicans ‘‘sent the White House a counter-offer that would achieve tax and entitlement reform to solve our looming debt crisis and create more American jobs.’’

Both sides say they want a deal to prevent damage to the economy, but that stated commitment has been accompanied by a fierce battle to gain the political high ground in negotiations — and the occasional comment that one side or the other would be willing to let the deadline pass without a deal unless it got acceptable terms.

Republicans acknowledge that Obama has an advantage in one respect, citing his re-election last month after a race in which he made higher taxes on the wealthy a centerpiece of his campaign.

At the same time, Republicans hold powerful leverage of their own, the certainty that by spring the president will be forced to ask Congress to raise the government’s borrowing authority. It was just such a threat that previously allowed them to extract $1 trillion in spending cuts from the White House and Democratic lawmakers, a situation that Obama has vowed he won’t let happen again.

In his noontime remarks on the House floor, Boehner said, ‘‘Let’s be honest. We’re broke. The plan we offered is consistent with the president’s call for a balanced approach.’’

‘‘We’re still waiting for the White House’’ to do the same,’’ added the Ohio Republican.

GOP senators across the Capitol soon echoed his remarks.

‘‘You have to ask the question, Is the president obsessed with raising taxes?’’ said Sen. John Thune of South Dakota, a member of the GOP leadership.

Referring to the president’s occasional outside-the-Beltway trips to build public support for his position, Thune said Obama was ‘‘doing a victory lap’’ after the campaign.

Senate Republican leader Mitch McConnell said GOP lawmakers are determined to overhaul benefit programs so they can ‘‘meet the demographics of the country.’’ He recently said Republicans want to curtail annual cost-of-living benefits for Social Security and other government benefits, as well as raise the age of eligibility for Medicare from 65 to 67 beginning at some point in the future.

‘‘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 on the Senate floor.