(CNN) Senate Majority Leader Mitch McConnell will delay the vote on the Republican leadership's health care bill until after the July 4 recess, two sources told CNN.

McConnell told GOP senators that he wants to make changes to the bill, get a new Congressional Budget Office score and have a vote after the holiday.

A White House official and a GOP aide on the Hill told CNN that White House chief of staff Reince Priebus and Vice President Mike Pence invited all Republican senators to White House on Tuesday afternoon.

Republican leadership, along with Pence, had sought earlier to woo members of their own party into supporting the fragile health care bill behind closed doors Tuesday, as a fifth senator came out against voting for a procedural step to advance their plans to repeal and replace Obamacare.

Mike Lee will vote against the motion to proceed on the health care bill as it is currently written, an aide told CNN, adding that Lee is "still negotiating with leadership to make changes as we have been since Saturday."

Walking through a hallway, Senate Majority Whip John Cornyn dropped the news that the Senate will hold a vote Wednesday on moving forward with the bill -- a normally routine procedural move that now looks grim due to Republican intraparty division.

With Republicans only able to lose two votes, the bill appeared to be teetering on the edge by lunchtime.

Sen. Richard Shelby, R-Alabama, told reporters the effort to get to 50 votes was "fluid" but maintained a degree of hope. "I wouldn't count McConnell out yet."

Meanwhile, Democrats on the Hill were organizing news conference after news conference, including a large showing of Democratic senators who gathered on the Capitol steps to lambast the already-beleaguered bill.

But back inside the Capitol, the full court press was on. Along with Pence, Priebus was also spotted on the Senate side.

Behind the scenes, staffers for McConnell were trading potential legislative proposals with key offices, and McConnell himself met privately with Sen. Ted Cruz for nearly an hour. While the Texas Republican hasn't said how he'll vote on the motion to proceed, he has been publicly opposed to the bill though open to negotiations.

Asked if he was still a "no" as he was leaving McConnell's office, Cruz told CNN: "It continues to be a work in progress."

Sen. Shelley Moore Capito, R-West Virginia, has been largely quiet since the bill went public last week. She's long been a question mark on how she'll vote on the bill, and she told CNN Tuesday she was "concerned about the bill in the form that it is now."

Asked later if she needed more opioid funding in the bill, Capito said that was just one of the issues. "That's not the only one," Capito told reporters.

Aides were seen scurrying around in the Capitol trying to buttonhole members and their key staff about where things currently stand, with the idea being to have a menu of options ready for the Senate Republicans' weekly Tuesday lunch, according to two GOP aides.

As McConnell and his top lieutenants attempt to thread the needle inside a wary conference, the lunch is expected to be at effort to both rally the troops and serve as a policy discussion. For his part, according to one aide, the message will be clear: A vote against the bill is a vote to maintain Obamacare. If the bill fails, McConnell will argue, the only option will be working with Democrats, which will certainly lead to a less conservative outcome.

While Cornyn offered a definitive timeline for the motion to proceed, Pence was less sure earlier. "We'll see," he told CNN when asked whether he thought the bill would be taken up on the floor.

Pressed on what they are doing to salvage the bill, he responded, "lots of good discussions underway and we'll continue to work very diligently."

Also at the Capitol was Florida Gov. Rick Scott, who was scheduled to hold meetings with Pence, McConnell and Sen. Marco Rubio, R-Florida.

Rubio, like some senators, has been waiting to make his decision on the bill until he feels like he has enough input from his state's governor and other constituents. An aide to the senator said Rubio's staff invited staff from the offices of Scott -- as well as the Florida speaker and Senate president -- to Washington to be "embedded" in Rubio's office for the week and work with Rubio's team to analyze how the bill will affect Florida.

Over on the House side, Speaker Paul Ryan was urging his members in a closed door session to give their Senate colleagues some space, according to a person in the room.

Ryan's point was that given the current state of play, it's not helpful for anyone in the House to come out and attack or criticize elements of the bill. Should the Senate pass the bill, Ryan argued, the House will get their chance, one way or another, to weigh in. Until then, he urged members to hold their fire.

CNN's MJ Lee, Lauren Fox, Ted Barrett and Phil Mattingly contributed to this report.