WASHINGTON – President Obama opened a private meeting with Senate Republicans on Thursday by asking them to search for compromise in the ongoing budget talks, but entrenched positions on both sides underscored the difficulty of resolving differences in time to avert a possible financial crisis.

According to Treasury Department calculations, Congress has until Aug. 2 to raise the legal ceiling — now $14.3 trillion — on the nation’s debt and avoid a default on government obligations.

Republicans have vowed not to raise the debt ceiling without agreement on far-reaching cuts in federal spending and no tax increases. Democrats have resisted some of the proposed cuts and called for increased federal revenues, saying the deficit cannot be reduced by spending cuts alone.

Thursday’s 90-minute exchange around a conference room table reflected a deepened divide over the role of government.

“We are talking here about spending reductions,” said Sen. Mitch McConnell of Kentucky, the Republican leader, in recounting his message to the president. “There will be no tax increases in connection with raising the debt ceiling.”

Broad agreement has emerged in Washington that chronic budget imbalances on the order of this year’s $1.5 trillion shortfall need to be fixed, which means the upcoming debt ceiling vote provides an opportunity for a budget deal since neither side will want to be blamed for a catastrophic default.

Yet even as both sides have found common ground on the broad goal of reducing deficits by $4 trillion over the next decade or so, stark differences in how to accomplish that goal remain.

On the way into the late-morning meeting, a White House aide stressed the “balanced approach” the president would preach — a reference to the need for both spending cuts and tax increases.

But McConnell countered in Thursday’s meeting by laying out the contours of a proposal to earn his vote, and presumably that of the other 46 Republican senators: Immediate budget caps over the next two years that press down discretionary spending, followed by gradual eligibility changes to the Medicare and Medicaid health programs. And no new taxes.