WASHINGTON — Hopes for a broad deficit-reduction agreement faded on Wednesday as President Obama insisted he had offered Republicans “a fair deal” while Speaker John A. Boehner moved for a House vote as early as Thursday on a scaled-down plan to limit tax increases to yearly incomes of $1 million and up, despite Senate opposition and Mr. Obama’s veto threat.

The impasse was clear as Mr. Obama and Mr. Boehner separately spoke to the television cameras instead of each other, after a weekend of private negotiations amid grieving over the shootings in Newtown, Conn., had narrowed their differences enough to raise optimism about a far-reaching deal to stabilize the nation’s debt.

First Mr. Obama and then Mr. Boehner faulted the other side for the impasse, and ultimately the failure, if a year-end deal could not be reached to stop automatic tax increases and the indiscriminate spending cuts in military and domestic programs known as the “sequester.” The president, saying he had gone “at least halfway” toward Republicans’ demands, evoked Hurricane Sandy and the Newtown school massacre to prod lawmakers to compromise for the nation’s benefit.

“When you think about what we’ve gone through over the last couple months — a devastating hurricane, and now one of the worst tragedies in our memory — the country deserves us to be willing to compromise on behalf of the greater good,” he said during an appearance at the White House to discuss gun control.