Washington (CNN) Four conservative Republican senators announced Thursday that they opposed the current version of GOP Senate leadership's health care bill as written.

Sens. Rand Paul of Kentucky, Ron Johnson of Wisconsin, Ted Cruz of Texas and Mike Lee of Utah said in a joint statement they're "not ready to vote for this bill."

"Currently, for a variety of reasons, we are not ready to vote for this bill, but we are open to negotiation and obtaining more information before it is brought to the floor," the senators said. "There are provisions in this draft that represent an improvement to our current health care system, but it does not appear this draft as written will accomplish the most important promise that we made to Americans: to repeal Obamacare and lower their health care costs."

Republicans can only lose two members of their 52-senator caucus in order to pass their proposal to repeal and replace Obamacare. In a 50-50 outcome, Vice President Mike Pence would provide the tie-breaking vote.

"As currently drafted, this bill draft does not do nearly enough to lower premiums," Cruz said in a separate statement obtained by CNN's Dana Bash. "That should be the central issue for Republicans -- repealing Obamacare and making healthcare more affordable. Because of this, I cannot support it as currently drafted, and I do not believe it has the votes to pass the Senate."

A number of other GOP senators are avoiding outright supporting the new health care bill unveiled by Republican leaders Thursday morning -- saying they need more time to read the fine print before taking a stand.

Senate Majority Leader Mitch McConnell unveiled GOP alternative to Obamacare, warning in a speech that "more Americans may get hurt if we don't do something."

"We agreed on the need to free Americans from Obamacare's mandates -- so Americans are no longer forced to buy insurance they don't need or can't afford," McConnell said on the floor. "We'll repeal the employer mandate so Americans no longer see their hours or take home pay cut because of it."

Despite McConnell's positive reinforcement, many Republican senators were not immediately sold on the draft legislation after a closed-door briefing Thursday morning.

Sen. Roger Wicker of Mississippi expressed some disappointment that the new bill still resembles Obamacare in some aspects.

"Clearly a lot of us would like to do more," he said. "We're dealing with reality and getting to 50."

Many scoffed at the short period of time they will have to read the bill -- one week -- before voting on it. McConnell said that the Congressional Budget Office scoring of the bill won't be released until next week. Sources tell CNN they are expecting it Monday or Tuesday.

"It'll be hard to pass the bill -- lot of differing points of view -- and one week may not be enough time," Senate Finance Chairman Orrin Hatch of Utah told CNN. Adding: "It's a real problem."

When asked if one week is enough time to review and amend the bill, GOP Sens. Bob Corker and Mike Rounds both responded "we'll see."

Members quickly rushed back to their offices to start reading the text of the bill. Many pointed to the tight timeline as their reason for avoiding questions about their thoughts.

Sen. Lisa Murkowski of Alaska -- a key swing vote -- said she's on her way back to her office to really go through the bill before she comments on it.

Asked what she thought about what she was told in the briefing she said there's "a lot there." Questioned if she had any insight on what the parliamentarian would do on abortion and she said she didn't know.

Corker said he's not "happy or unhappy" about the bill. that he's going to go back and read it.

Sen. Bill Cassidy of Louisiana echoed the same refrain, saying he wants to read the bill in full before articulating his concerns about it.

Earlier Cassidy had spoken about the bill needing to meet the Jimmy Kimmel test, harkening back to that viral monologue where the late night comedian revealed his son's heart condition and called out Republicans for considering a new bill that would allow insurance companies to drop coverage or charge more for pre-existing conditions.

Asked by CNN if the new bill met his "Kimmel test," Cassidy responded, "Again, I need to review the text but the issue is if your loved one gets sick, would they have adequate coverage? As best as I can tell it does but again I need to read the text."

Sen. John McCain, R-Arizona, generally praised the bill, and said it was better than Obamacare in "100 ways."

However, he also said he still needed more time to study it before deciding if he'd support it, and in particular wanted feedback from his home state's governor on Medicaid provisions.

Nevada Sen. Dean Heller said he has "serious concerns" and also said he wants to see what his state's governor says.

"At first glance, I have serious concerns about the bill's impact on the Nevadans who depend on Medicaid," Heller said.

Ready to vote?

The short turnaround begs the question of whether one week really is enough time to vote on a bill that could restructure 1/6th of the economy, and if Republicans will wait until next month.

However, South Carolina Sen. Lindsay Graham said "I think so" when asked whether there is enough time to debate the bill.

McCain said he wasn't happy with this process but thought one week was enough time to review the bill for a vote.

Graham cautioned though, that it's likely that the bill won't stay in its current form.

"I think it'll probably get a lot of push back from people on the right in the house," he said adding that he himself came away more "positive" than he thought he'd be.

Graham expanded, "This bill is not take it or leave it. It is a starting point where we can look at the bill from our state perspectives."