President Barack Obama made a budget counteroffer to Speaker John A. Boehner on Monday that included raising the top rates on income above $400,000, $400 billion in health care cuts and $200 billion in discretionary cuts.

Sources familiar with the talks between the White House and the Ohio Republican confirmed the new offer, which no one characterized as close to final. However, it marks progress as lawmakers hurtle toward the Christmas holiday and their Jan. 1 deadline. Without action, taxes are expected to go up on all Americans and deep automatic spending cuts are scheduled to begin.

House Republicans rejected the offer but noted it was “a step in the right direction.”

One source characterized the offer as including $1.2 trillion in cuts, including almost $300 billion in saved interest costs, while raising $1.2 trillion in revenue, down from $1.4 trillion in an earlier offer and $1.6 trillion in the administration’s original ask.

Boehner’s office, however, disputed those figures, unwilling to give credit for interest savings as a cut and saying the net revenue would be closer to $1.3 trillion because of effects from the changes in inflation calculations on the tax code.

“Any movement away from the unrealistic offers the president has made previously is a step in the right direction, but a proposal that includes $1.3 trillion in revenue for only $930 billion in spending cuts cannot be considered balanced,” Boehner spokesman Brendan Buck said. “We hope to continue discussions with the president so we can reach an agreement that is truly balanced and begins to solve our spending problem.”

Boehner recently offered to increase the GOP’s revenue offer to $1 trillion from $800 billion — the same level as from his last negotiations with Obama. Obama’s offer of the $400,000 income threshold is higher than his initial position of $250,000.

The White House proposal, first reported by The Associated Press, also includes a debt limit increase for two years, a permanent extension of expired tax extenders and a turning off “except in select areas” of the sequester — the automatic across-the-board cuts triggered by the failure of the deficit reduction supercommittee to come to a $1.2 trillion budget deal in 2011.

In his first offer to Obama, Boehner proposed changes to the way the Consumer Price Index is calculated. By using chained CPI, Boehner estimated savings of $200 billion over a decade. The Moment of Truth Project estimates that chained CPI, if fully implemented to reduce the growth of cost of living increases in Social Security and federal retirement programs as well as raise additional revenue, would save $236 billion over a decade. The White House offer, however, refers to a “superlative” CPI, which would provide protection to the “most vulnerable” populations receiving federal benefits and would save somewhat less — $130 billion over 10 years.

The plan would eliminate at least most of the sequester. Part of the sequester could still take effect as a mechanism to enforce the agreement, a source said.

It also provides for spending on infrastructure and extending unemployment insurance but apparently would not extend the temporary payroll tax cut.

Under the plan, the debt ceiling would be extended for two years, twice as long as Boehner is reported to have proposed in an offer to Obama last week.

The proposal would extend provisions that prevent the alternative minimum tax from hitting millions more taxpayers, as well as prevent cuts in federal payments to doctors who treat Medicare patients.

The Speaker met Monday evening with his GOP leadership team and key players on budget issues — including former vice presidential nominee Paul Ryan of Wisconsin — but no one exiting Boehner’s office suite would comment on the internal conversations.

The $2.4 trillion proposal, when added to the existing $1.1 trillion in spending cuts from the Budget Control Act and war savings, would produce $4 trillion in deficit reduction, according to some sources.

Paul Krawzak contributed to this report.