Although members of his own party show signs of impatience with some administration policies, President Bush is counting on House Republicans to back him in an appropriations showdown.

To prepare for battle over spending bills this summer and fall, the White House is nurturing relationships that were strained by a split over the minimum wage and, more recently, by the immigration debate.

Bush and House GOP leaders want to back Democrats into a corner where they will have to pare back their spending bills or risk being on the losing side of veto override votes. If the GOP stands firm, the result could be one big fight over an omnibus spending package after a series of battles over individual appropriations measures.

The key number is 145.

With all 435 House members voting, the White House would need 145 votes to prevent its opponents from mustering a two-thirds majority to override a veto.

There are three vacancies; all are expected to be filled in the next few months.

There are 201 House Republicans. Bush, aided by a revamped White House lobbying team with ties to the House GOP, has to convince at least 145 that backing his position on spending would have a bigger political payoff than would the local goodies in vetoed funding bills.

Already, 147 House Republicans have signed a pledge to sustain Bush’s threatened vetoes of spending bills that exceed his budget requests for those bills.

“Our strategy on spending will go hand-in-hand with [Bush’s] veto pen,’’ said Minority Whip Roy Blunt, R-Mo. “We’ve maintained a veto-sustaining number on every appropriations bill up until now.”

The House passed its version of the Interior spending bill (HR 2643) on a vote of 272-155; the State-Foreign Operations bill (HR 2764) on a vote of 241-178, and the Homeland Security (HR 2638) bill by a margin of 268-150. Both the Interior and Homeland Security bills would exceed the president’s requested spending level.

The White House has threatened vetoes of all three bills, as well as the House Energy and Water (HR 2641) appropriations bill. Another veto threat is expected on the Defense appropriations bill, to which Democrats hope to add language calling for withdrawal from Iraq.

Democrats have yet to form a plan for handling vetoes — beyond accusing the GOP of obstruction. “People are frustrated with the fact that things haven’t changed more quickly. But they understand it’s the Republicans that are standing in the way,” said Chris Van Hollen of Maryland, chairman of the Democratic Congressional Campaign Committee.

GOP leaders say they are confident they can rally Republicans to sustain vetoes on most, if not all, of those bills, and both sides predict the spending battle will extend well into the fall.

There is bound to be pressure on the 147 pledge-signers to stick with Bush on the difficult spending decisions ahead — and even more pressure on the 54 House Republicans who have not signed. Many in the latter group — including Wayne T. Gilchrest of Maryland, Jerry Lewis of California and Phil English of Pennsylvania — are members of the centrist Republican Main Street Partnership.

On July 4, Bush reached out to one Main Street member who has not yet pledged to uphold spending vetoes: West Virginia’s Shelley Moore Capito. The president joined her at an Air National Guard hangar in Martinsburg, where he praised her patriotism, saying, “When we’ve got somebody in harm’s way, she understands what I understand — that military person and his or her family deserve the very strongest support from the federal government at all times.”

That kind of overture is likely to be repeated in other districts this summer.

Bush remains an asset in GOP districts, said Tom Cole of Oklahoma, chairman of the National Republican Congressional Committee. “Drawing a line on spending will be good for the president and good for Republicans,’’ he said.

After party-splitting fights on immigration (S 1639) and increasing the minimum wage to $7.25 (PL 110-28), GOP leaders decided during a June 26 meeting with Bush to concentrate on areas of agreement: opposing spending they view as excessive and blocking efforts to increase taxes that oil and gas companies pay (HR 2776).

“We are reuniting around . . . core principles to keep us together as a party,” said House Minority Leader John A. Boehner, R-Ohio. The GOP rank-and-file, he reported, wants to work with Bush to “home in on . . . principles that frankly got a little lost over the last few years.”

But former Rep. Bill Frenzel, R-Minn. (1971-91), a Brookings Institution scholar, said the White House and Boehner may have trouble keeping House Republicans together when what’s best for the White House may not help their re-election campaigns.

Voting to sustain vetoes of appropriations bills often means voting against local money. “That’s going to be harder to do as time goes on,” Frenzel said.

Any victories that Bush rings up on spending bill vetoes with the help of House Republicans will not guarantee the party unity extends to other issues.

For instance, Blunt wants the House GOP to oppose reauthorization of the No Child Left Behind education law (PL 107-110) on grounds it undermines state and local authority. Boehner helped write that law with Education and Labor Chairman George Miller, D-Calif. Boehner and Bush want it renewed. The rest of the GOP Conference hasn’t had to weigh in yet.