Story highlights 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.

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