Asked whether he will move forward with or without the necessary votes, he said he doesn't expect that to be an issue.
"Well, I expect to have the support to get it done," Cornyn, the senior senator from Texas, told ABC News congressional correspondent Mary Bruce. "And yes, we will vote this week."
The vote in question is a procedural one to allow Senate Majority Leader Mitch McConnell to bring the bill to the floor and open it up for discussion before a final vote on passage.
"We're still talking to a few who have concerns," Cornyn said. "My optimism comes from the fact that most of them are saying that we want to get to yes and we just have some concerns. Obviously, the discussion draft that was released was just that. And it's produced some discussions. A lot of the news among senators — we are continuing to work on it. And I remain optimistic.
"I'm not willing to accept that we are not going to be successful, because I think the consequences are very bad."
In order to pass the health care bill through the Senate, Republicans need to hold on to 50 of their 52 votes. They can afford only two defections; in case of a tie, they have the option of calling in Vice President Mike Pence to cast the tie-breaking vote. But because the latest count has at least five Republicans saying they will vote against allowing debate over the bill to proceed, a final vote may not even take place.
Pence is playing a role in the fate of the health care bill in a different way as well. He is having dinner with four of the holdouts tonight in an effort to talk through some of their concerns and decide whether he can sway them in favor of the bill.
Paul is reportedly meeting with President Trump at the White House today to talk about the bill, according to Paul's office. Paul has been one of the most ardent and outspoken Republicans opposed to the bill.
Breaking down the CBO numbers
Before the Senate may begin debate on most legislation, the senators must unanimously agree to consider it, or the majority leader must offer a motion to proceed to consideration.
The 22 million figure, which is only a slight improvement from the CBO's estimate of the health care bill passed by the House of Representatives in May, comes in the office's analysis of the Better Care Reconciliation Act, a draft of which was released last week.
The act, which faces staunch opposition from Democrats, could further result in a reduction of the cumulative federal deficit by $321 billion by 2026, largely due to cuts in Medicaid spending, according to the CBO report.
"I won't vote to proceed unless the bill changes," Paul told reporters Monday. He wants the GOP legislation to go even further in rolling back certain Affordable Care Act measures.
"I had a long conversation with the president last night, and I think he's open to negotiations, but we have not had any word from anyone in Senate leadership," Paul said.
"No one from leadership has reached out to us," he added.
"I would highly doubt I would support it," Johnson told reporters of the motion to proceed. He has been adamant in recent days about holding off on a vote on the bill — which GOP leadership is pushing to take place before Congress' July 4 recess.
Through a spokesman, Lee confirmed to ABC News that he also intends to vote "no" on the procedural vote unless changes are made.
On Friday at a press conference with the Nevada governor, Heller said he would not support the bill as is.
Is there room to negotiate?
Moments after the CBO score was released, Sen. Lindsay Graham, R-S.C., said the vote was getting tougher by the minute.
"How you put all this together and get to 50 is going to be very tough, and the CBO score doesn't help any. If you had problems with the bill before, you're probably going to have more problems now," he said.
Graham said if senators are considering voting "no" because of how it would affect their states, then they should stick to it and vote "no."
"I don't believe in this running off a cliff, like our Democratic friends did. They got this herd mentality where we got to pass this bill or else," he said.
Graham is leaning toward voting "yes" on the bill.
ABC News' MaryAlice Parks and Adam Kelsey contributed to this report.