President Obama delivered Speaker John A. Boehner a new offer on Monday to resolve the pending fiscal crisis — and what may be close to a final deal, which would raise revenues by $1.2 trillion over the next decade but keep in place the Bush-era tax rates for any household with earnings below $400,000.

The offer is close to the plan that Mr. Boehner proposed on Friday, according to officials familiar with it. Other officials say the speaker will present the plan to House Republicans Tuesday morning, and both sides are cautiously optimistic that a major deficit reduction plan could be passed well before January, when more than a half-trillion dollars in automatic tax increases and spending cuts would kick in.