By David Lawder

WASHINGTON (Reuters) - Republicans in the House of Representatives pressed ahead on Tuesday with efforts to extend deep "sequester" spending cuts into next year, despite a veto threat from President Barack Obama.

With no progress toward reconciling deeply divided House and Senate budget resolutions and the deadline to raise the federal borrowing limit fading well into the fall, Congress' battle over deficit reduction is shifting to the spending bills to fund government agencies and programs for the fiscal year starting on October 1

The Republican-led House on Tuesday formally adopted a sharply reduced, $967 billion discretionary spending cap and passed the first appropriations bill written to that level, a measure to fund military construction and the Veterans Administration.

The $73 billion bill, which gets a $2.4 billion boost over current sequester levels, passed with overwhelming bipartisan support, 421-4, as Democrats chose to demonstrate support for U.S. veterans.

The House Appropriations Committee also unveiled two other bills that conform to the sequester-induced cap - $512.5 billion in Defense Department non-war funding, and $19.5 billion for the Department of Agriculture.

But the Pentagon bill gets a $28.1 billion boost above current sequester levels, which will require that funds be shifted from domestic agencies. Education, medical research and other programs will face deeper cuts below their sequester-induced levels.

The discretionary spending levels set forth by House Republicans assume that sequester cuts continue. But the Democratic-controlled Senate is assuming that the cuts get replaced with savings elsewhere and is working toward spending bills written to a higher $1.058 trillion level.

"It is our duty and responsibility to prioritize tax dollars and target funding to where it's needed most, and there is absolutely no higher national priority than the protection of our country through a strong national defense," said House Appropriations Chairman Hal Rogers, a Republican from Kentucky.

Later on the House floor, Rogers said he was still hoping for a budget deal that would allow the House and Senate to restore some discretionary spending.

Commenting on the House's adoption of the Republican spending cap, the No. 2 Democrat in the chamber, Steny Hoyer of Maryland, said: "So what's before us is before us: a ratification of sequester, which starts with ‘S,' which stands for stupid. It is a terrible process."

Unless replaced with other spending cuts or tax increases, the sequester will shrink spending across the board by about another $1 trillion over a decade. It was created by a 2011 budget deal and took effect in March after Congress failed to find alternate savings.

SPENDING STOPGAP SEEN LIKELY

On Monday, the White House budget office said it would recommend that Obama veto any House spending bills that implemented what it called "draconian" budget cuts and that were not passed in the context of a broader budget deal.

Obama is unlikely to have to make good on the threat because the deep differences between the House and Senate spending approaches mean that Congress will struggle to pass appropriations bills in time for the new fiscal year.

A senior Republican aide said another stopgap funding bill would likely to be needed to avoid a government shutdown on October 1.

Senate Democrats, who passed their first budget in four years in March, have been calling on House Speaker John Boehner to appoint conferees to work out differences between the Democratic and Republican fiscal visions. But Republicans have declined, arguing that the two sides are too far apart to begin public conference meetings that would likely end in failure.

(Reporting by David Lawder; Editing by Peter Cooney)