– House Republicans journeyed to the White House Thursday for a health care victory lap, but Senate Republicans were in no mood for celebration.

Instead, they sent an unmistakable message: When it comes to health care, we’re going to do our own thing.

“I think there will be essentially a Senate bill,” said Sen. Roy Blunt, R-Mo., the fourth-ranking Senate Republican.

Now that the House has passed legislation overhauling the nation’s health care system, it is headed to the Senate, where Republican leaders will wrestle with challenges complicating its chances.

Republican senators are signaling that their strategy will be rooted in crafting their own replacement for the Affordable Care Act. It remains to be seen how closely that measure will resemble the one that passed the House on Thursday.

A group of GOP senators met Thursday in the office of Senate Majority Leader Mitch McConnell, R-Ky., to begin outlining their priorities, said Sen. John Cornyn, R-Texas.

“It was designed by the leader to be a smaller group of people that represent the different perspectives and points of view in our conference,” Cornyn said. “If that group can get to yes, then we will take it to the rest of the conference.”

But Cornyn would not commit to a timeline for a Senate vote, simply saying: “When we get 51 senators, we’ll vote.”

Republicans hold a 52 to 48 edge over Democrats in the upper chamber, leaving them a narrower margin for error.

In a sign of the frustration that some GOP senators have with the House bill, Sen. Lindsey Graham, R-S.C., posted a skeptical note on Twitter Thursday: “A bill — finalized yesterday, has not been scored, amendments not allowed, and 3 hours final debate — should be viewed with caution.”

Senate Republicans have opted to use a maneuver known as reconciliation to try to pass the bill with a simple majority, instead of having to clear the 60-vote threshold that is required for most legislation. In the current balance of power, that would require Democratic votes. But even getting to a simple majority will be no small task.

GOP senators from states that have expanded Medicaid under the Affordable Care Act, such as Sen. Rob Portman of Ohio and Shelley Moore Capito of West Virginia, have concerns about rollbacks to that program in the House bill.

Meanwhile, a trio of conservative senators — Ted Cruz, R-Texas; Mike Lee, R-Utah, and Rand Paul, R-Ky. — are also wild cards. Earlier this year, they pushed for a more aggressive repeal of the health care law than many of their colleagues favored.

The House voted before they received a score from the nonpartisan Congressional Budget Office, which measures how much the legislation would cost and how many people stand to lose coverage under it. Senate rules require a CBO score that proves the legislation will not increase the deficit after 10 years. A score from the CBO is expected to take several weeks.

The CBO said in March that an earlier version of the GOP health care plan would result in 14 million more people being uninsured in 2018. It projected that the plan would cut the deficit by $150 billion between 2017 and 2026.