Boehner has offered about $1 trillion in tax increases and roughly the same amount in spending savings. An earlier Boehner offer included $600 billion in Medicare and Medicaid savings — well more than Obama — but it’s unclear whether the speaker is still seeking that figure.

Because of a dispute over how some savings are classified, Boehner says Obama’s offer is really $1.3 trillion in higher taxes and only about $850 billion in spending cuts.

The House speaker says Obama’s offer is not balanced because its new taxes and spending cuts are unequal. And he complains it does too little to control fast-growing benefit programs like Medicare, a chief driver of the federal government’s mushrooming deficits.

‘‘The real issue here, as we all know, is spending,’’ Boehner said Thursday. ‘‘You go through all these discussions, I don’t think the White House has gotten serious about the big spending problem the country faces.’’

The two men’s differences work out to $200 billion over 10 years in taxes, and about the same in spending, depending on whose numbers are used. Either way, their gap is less than 1 percent of the money the government will spend and tax anyway.

‘‘They’re a couple hundred billion apart. This is absolutely senseless,’’ said Senate Majority Leader Harry Reid, D-Nev., insisting Boehner should compromise. ‘‘These are gyrations I've never seen before.’’

There are other differences, too.

Obama wants several billion dollars in infrastructure spending to goose the economy and to extend expiring unemployment benefits. He also wants the government’s authority to borrow money extended for two more years — until after the 2014 congressional elections — with Congress having little more than symbolic opportunities to block it, a year longer than Boehner has offered.

Even so, the numbers being proposed by Obama and Boehner are so close, and the political risks both men have taken on taxes and Social Security benefits are so stark, that many consider it almost unthinkable that they would not eventually complete a deal.

‘‘Having come out of their trenches, they either have to shake hands or get shot, maybe by their own troops,’’ said Robert Bixby, executive director of the nonpartisan Concord Coalition, an anti-deficit group.

___