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Senate Republicans have yet to unveil their measure

"I'll let Sen. McConnell determine the Senate schedule," Spicer said

CNN  — 

Donald Trump may be the President of the United States, but when it comes to passing health care reform through the Senate, Trump is the understudy to Mitch McConnell’s lead.

Senators and their top aides on Capitol Hill have made it clear to the White House as health care legislation makes its way through the Senate that the less involved Trump is, the better for the bill’s prospects.

It’s a tangible acknowledgment that Trump’s pull with Senate Republicans, many of whom were not eager to back his presidential campaign, is far weaker than with House Republicans.

“The White House has been super hands off, which feels just about right,” a Republican aide told CNN.

It also reflects a strategy substantiated earlier this month during Trump’s most public foray into the health care process: Hosting an eclectic group of senators for a White House policy luncheon.

During the public portion of the event – when reporters were invited in – Trump undercut the House bill, saying more funding was needed to make it “generous, kind (and) with heart.” Behind closed doors, he also called the measure “mean,” CNN reported.

The comment highlighted Trump’s legislative naivete: The bill passed through the body has to save $133 billion, the same amount of money as the House bill, in order to be passed with a straight majority. That leaves Trump and Senate Republicans with little flexibility on spending. While they can spend more money on certain areas, they still have to hit the $133 billion marker.

The comment sent shock waves through Capitol Hill. Democrats have used it to rally their base against the GOP-crafted measure, while Republicans have begun to question the value in going out on a limb for a President who mere days after publicly cheering a bill would trash it in private.

How in the dark is the President? With a possible vote days away, the White House hasn’t seen the health care bill in its entirety, according to a senior White House official. This official said the White House has provided “technical assistance” to senators in crafting it.

“Everybody’s going to see it in entirety real soon,” the White House official told CNN.

The strategy is an acknowledgment of Trump’s lack of policy prowess, and White House aides are aware of that.

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McConnell, the Senate majority leader, is viewed inside the White House and around Washington as a master of the Senate, someone who knows his caucus and what it will take to get a bill passed. Trump, it’s widely acknowledged, is not.

But White House top aides also see the distance as a way to inoculate Trump from considerable blame if the bill fails, a nod to the President’s low approval rating and waning political capital.

“No one knows the Senate better than McConnell,” one White House official said when asked about Trump’s involvement, an acknowledgment that he is taking the lead.

A Republican Senate aide agreed.

“This is a leadership bill, sink or swim. All credit and all blame stays in the Senate,” the aide said, adding this political reality: “This gives the President cover, in a sense, if it fails.”

Trump, however, will received plenty of blame if the bill fails, whether his involvement is kept to a minimum or not. Just as, if the bill succeeds, Trump will surely be the first to spike the football, much like he did in the Rose Garden after health care reform narrowly passed the House.

So far, the White House’s embrace of Senate strategy shows some growth from the five-month-old administration. During the debate over health care in the House, White House aides were eager to set deadline, dates and expectations for the bill, comments that angered even staunch Trump supporters in the House.

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White House press secretary Sean Spicer did the opposite on Tuesday.

“I’ll let Sen. McConnell determine the Senate schedule and run the Senate that he sees fit,” Spicer said.

Vice President Mike Pence, a former congressman, has been more involved than Trump with the Senate debate, regularly chatting with lawmakers, including meetings on Tuesday.

Pence, who has substantially more legislative experience than Trump, has attend a luncheon with Senate Republicans nearly every week, checking in on the health care process as a liaison for the White House.

CNN’s Elizabeth Landers contributed to this report