(CNN) For months, House Republicans agonized over how to pass a bill to repeal Obamacare. Now it's the Senate's turn to feel the pressure.

Republican House members are trying to give Senators room to negotiate their own bill, but it's a delicate balance. Once the Senate bill passes, House Republicans may be expected to weigh in quickly. They are caught between wanting to stake out clearly what they cannot support without interjecting too much into what is a fluid negotiation.

"We'll wait and see what comes back," said Ohio Republican Rep. Jim Jordan, a member of the House Freedom Caucus.

The timing of when the Senate's bill -- assuming it passes late next week -- could come up in the House is still an open question. Democrats in the House are on alert in case Republican leaders move quickly to push the Senate bill through the House Chamber before the July 4 recess. The House is expected to be in session next Friday.

However, Oklahoma Republican Rep. Tom Cole, a member of the House whip team, was skeptical a health care bill would come to the House floor that quickly.

"I don't think so," Cole said when asked about a Friday vote. "I don't see how we could do that. Most of our members have not had a chance to really look at the Senate bill assuming it would pass."

Many members of the House conference told CNN Friday that they were still reviewing the Senate's legislation and weren't quite ready to jump to any conclusions about whether they could ultimately support it.

"I think for us, we really didn't appreciate how obnoxious senators were," a House GOP aide said about their own process. "Ultimately our goal is to get this thing to get passed so we aren't going to muck up the process. The last thing you need is your own friends taking digs at you."

Members are trying to give the Senate room to work, but that doesn't mean House conservatives aren't trying to put down a few markers. Conservatives worry the Senate bill doesn't go far enough to repeal Obamacare regulations that drive up the costs of premiums. They're frustrated the Senate bill keeps in place Medicaid expansion longer than theirs did.

"As of right now, the Senate bill does not have enough conservative support to get through either chamber to President Trump's desk," House Freedom Caucus Chairman Mark Meadows told CNN. "There will need to be changes, but I'm optimistic that we can work with the Senate to get those changes and ultimately work something out."

House Republicans are also fearful that under Senate rules, key anti-abortion provisions in the Senate bill may be ruled out of order and will be taken out.

"The thing that is obviously unwritten reality is that they're having to do this in the Senate within the confines of the Byrd rule, which may threaten even the pro-life stuff. If they do that, they blow this thing to Mars," Arizona Republican Rep. Trent Franks said, referencing the Byrd rule, which is the rule by which senators can pass bills under reconciliation.

Under Senate reconciliation -- the process that allows the Senate to pass a bill with just 51 votes -- Senate Republicans are more limited than they were in the House as to what policies can be considered.

One of those provisions that could be on the chopping block would be a one-year defund of Planned Parenthood. The provision survived the so-called Byrd rule in 2015, but there has been some speculation it could be ruled out of order.

That has members on edge.

"The Planned Parenthood (defunding) can't come out" if the Senate expects House conservatives to remain on board, Jordan said: "That's for darn sure."

But it's not just House conservatives who are anxiously awaiting the Senate negotiation over the next week. Many House moderates walked the plank for leadership in May and voted for the House version of Obamacare repeal even though they had serious reservations. They're hoping the Senate bill is more moderate, passes and comes back to them for a vote. If not, they took a tough vote that could cost them in their re-elections without an actual law to show for their gamble.

Still, other moderates who didn't vote for the House bill say they don't think the Senate bill goes far enough to assuage their concerns.

"I'm doing due diligence on it now, but it presses me as being similar to the House bill in many ways and I did not vote for the House bill," said Rep. Leonard Lance, a New Jersey Republican. "I have several, very serious concerns."