President Bush will plumb the depth of his clout on Capitol Hill when he asks Senate Republicans to resurrect the stalled immigration bill that is his chief domestic priority.

Bush’s personal intervention Tuesday is not expected to win many hearts, minds or votes for the immigration bill (S 1348), according to Republican lawmakers, lobbyists and aides.

“We’d be interested in the president’s advice, but I think this is not an issue upon which many people are undecided,” Senate Minority Leader Mitch McConnell, R-Ky., said Monday.

Bush once steamrolled critics in his own party on issues ranging from trade to the Medicare prescription-drug law (PL 108-173). But electoral losses and voter discontent with the management of the Iraq War and other issues have taken their toll, and the White House has to work harder than ever to keep the president’s party behind him — as the defections of GOP senators demonstrated during Monday’s procedural vote on a resolution expressing no confidence in Bush’s attorney general (S J Res 14). (See Gonzales, p. 39)

For the immigration overhaul, the best Bush may be able to do is stem the erosion of the measure’s bipartisan coalition.

“I’m not sure he’s coming here to change minds,” said Senate Republican Conference Chairman Jon Kyl, R-Ariz., who is one of the bill’s authors. “He’s trying to galvanize support.”

Several supporters said they hoped Bush’s appearance Tuesday would breathe life into the bill’s prospects.

But even backers say Bush’s sway is limited on immigration because so many in the GOP believe the bill would give “amnesty” to illegal immigrants. And the president’s clout with Congress overall is not what it used to be, Republican lawmakers and lobbyists say privately.

“They don’t feel like they owe him anything. They feel like he’s weighing them down, and it’s every man, woman and child for himself,” said a Republican lobbyist who is close to Senate GOP leaders.

Noting the adulation Bush enjoyed on his recent trip to former communist countries, the lobbyist said Bush may “be wishing the Albanians were his base instead of the Republican Conference.”

The immigration issue echoes debates of the past in which the administration failed to win support from congressional Republicans for domestic policies, most notably Bush’s plan to overhaul the Social Security system with private accounts in the 109th Congress.

But the emotional nature of the immigration debate — and Bush’s willingness to embrace some of the same concepts as liberal Democrats — have put him at an even greater disadvantage in lobbying fellow Republicans, lawmakers and aides say.

Clearly aware of Bush’s vulnerability on the issue, Senate Majority Leader Harry Reid, D-Nev., wrote Bush a letter Monday asking him to ratchet up his campaign for the immigration overhaul.

“We appreciate the efforts of you and other Republicans who have worked with us to get the bill this far,” Reid wrote. “But it will take stronger leadership by you to ensure that opponents of the bill do not block its path forward.”

Most Republican lawmakers said Bush still carries some influence, and some insisted he can make a difference on immigration.

“I thought it was dead. Now it’s on life-support, and there’s a good surgeon on hand,” Rep. Ileana Ros-Lehtinen, R-Fla., said of Bush. “There’s hope. I would not pull the plug.”

But even backers of the immigration bill had a hard time identifying a single senator whose vote the president could win.

Sen. Wayne Allard, R-Colo., said he is still undecided about the bill but declined to say that his vote could be won by the president.

“I will listen,” Allard said.

Republicans on Capitol Hill are not eager to publicly criticize their party’s chief executive, but they will describe at length their frustrations with what they see as a White House that has preferred to give instruction to Congress rather than seek its input.

That dynamic has been evident in the waning Republican support for Attorney General Alberto R. Gonzales.

Gonzales was spared the indignity of a “no confidence” vote when Senate Democrats failed to muster the 60 supporters needed to end debate and move to a roll call Monday. But seven Senate Republicans, five of whom are up for re-election in 2008, abandoned the nation’s top law enforcement official.

“Why is Gonzales really in trouble? He never groomed relationships,” said Rep. Jack Kingston, a Georgia Republican. “He never made any friends over here.”

Likewise, the Republican lobbyist said, “Even if [Bush]were at the top of his game, he would have a hard time moving anyone” on immigration.

“Given his standing in the polls and the lingering resentment among Republicans about the last election, I think there’s going to be hardly any sway at all,” the lobbyist said.