(CNN) Congress returns Monday from its Fourth of July break to face a critical three-week sprint during which Republican leaders hope to resolve key stalled issues before leaving town again at the end of the month for the long August recess.

The most pressing issue for Republicans is to put some momentum behind a deal on health care, which appears to have stalled during their recess. Congress also faces thorny talks on the budget and taxes, a hearing for President Donald Trump's nominee to replace James Comey as head of the FBI, and what to do about Russian sanctions.

Health bill in trouble

The top priority for Senate Republicans is to find a deal on health care reform, elusive until now -- and possibly out of reach -- because of the deep ideological and policy divides between GOP moderates and conservatives.

Trump highlighted the importance of the issue first thing Monday morning on Twitter.

"I cannot imagine that Congress would dare to leave Washington without a beautiful new HealthCare bill fully approved and ready to go!" Trump tweeted just before 7 a.m. ET.

I cannot imagine that Congress would dare to leave Washington without a beautiful new HealthCare bill fully approved and ready to go! — Donald J. Trump (@realDonaldTrump) July 10, 2017

Those dug-in corners of the caucus disagree on several issues, including how robust insurance regulations should be, how quickly to phase out the Affordable Care Act's expansion of Medicaid and how deeply to slash funding for the program, which is currently providing health coverage to more than 70 million Americans or nearly one-fifth of the country.

Right now, there is no Senate vote scheduled on this long-sought Obamacare repeal and replace bill and it's not clear when or if Senate Majority Leader Mitch McConnell will call one.

Vice President Mike Pence, who's been a key voice in the health care discussions, went horse back riding Saturday with Sen. Roy Blunt, the vice chairman of the Senate Republican Conference, as well as another crucial official in the health care debate: Seema Verma , the administrator for the Centers for Medicare and Medicaid Services.

Thanks @RoyBlunt for joining @SeemaCMS @SecretaryZinke & me on ride thru Rock Creek Park & thanks to @usparkpolicepio for your work everyday pic.twitter.com/rWSM0Gyznh — Vice President Pence (@VP) July 8, 2017

McConnell further fueled speculation that the bill is doomed when he told a hometown audience that if a deal cannot be reached soon, he would move to shore up struggling private insurance markets under Obamacare, something that would require cooperation with Democrats to achieve. But those comments are also being viewed as a warning shot -- a wake-up call of sorts -- to fellow Republicans: They can vote for an imperfect bill or give up on their years-long quest to gut Obamacare.

One of the proposals the CBO is examining was written by Sen. Ted Cruz, a Texas Republican, a key player in the talks who currently opposes the bill. It would allow insurance companies to offer inexpensive, bare-bones plans in an effort to drive down premiums, a top priority for the conservatives. But if it's included in a final bill, it could push away moderates and make it impossible for a bill to pass. Moderates worry that the Cruz amendment would lead to premium hikes for the sick, weaken protections for those with pre-existing conditions and cause insurers to flee.

"I think there's very little interest in the caucus in touching pre-existing conditions, so I have a hard time seeing the addition of the 'Consumer Choice Amendment,'" said one GOP Senate aide, referencing the formal name of Cruz's amendment. "And outside health policy folks have said that would set up a death spiral for the markets."

Sen. John McCain did not sound very optimistic about his party's chances for repealing and replacing Obamacare when he described Sunday the Senate proposal as "probably going to be dead."

"I fear that it's going to fail, and then we should convene a Republican Conference and say, 'what are we going to do'?" McCain said on CBS' "Face the Nation." "Introduce a bill. Say to the Democrats, 'here's a bill.' It doesn't mean they don't -- that they control it. It means they can have amendments considered. And even when they lose, then they're part of the process. That's what democracy is supposed to be all about."

McCain argued Republicans risk repeating the mistakes Democrats made when they passed the Affordable Care Act without any votes from the opposing party.

"Only, guess what?" he asked moderator John Dickerson. "We don't have 60 votes, John."

Key GOP meeting Tuesday

McConnell recently colorfully compared finding a path to 51 votes in the Senate to trying to solve a Rubik's Cube. And with about 10 GOP senators holding back their support, McConnell's challenge remains three-dimensional.

A key event to watch this week will be the Republican policy lunch Tuesday in the Capitol when McConnell and other GOP leaders may lay out for their wary rank-and-file members any emerging compromise to see if they can support it.

He delayed the vote until after the July 4 break. Now, some Republicans are calling for the August recess to be canceled if a deal isn't reached by then. And it's not clear McConnell will want to drag the issue out any longer. He also has other key issues demanding the Senate's attention -- including government funding, a debt ceiling increase and tax reform, a top priority of the GOP Congress and the White House.

Spending, taxes, debt

In the House, GOP leaders are trying to unlock stalled budget talks. Again, moderates and conservatives are clashing over spending levels and priorities. Conservatives want more money for defense and to slash spending on domestic programs and entitlements. But moderates are digging in to protect many of those programs.

Speaker Paul Ryan said Friday he's hopeful of getting a budget passed before the August break.

"I would prefer to pass one in July before we have to get all of our fall work done," he told reporters. "So, that's my goal, my preference. We're going to get back with our members starting Tuesday to talk about that."

The outcome could be critical to the legislative agenda for congressional Republicans and President Donald Trump because if a budget resolution is approved by the House and Senate, Republicans will be able to use special budget rules to pass a tax overhaul without the threat of a filibuster from Senate Democrats. Republicans currently are attempting to use those same rules to pass the health care bill, evidence that passing a tax a bill could prove just as tricky.

One pressing issue Congress won't need to deal with until the fall is raising the debt limit. The Congressional Budget Office announced recently it can keep paying America's bills through October. That will give lawmakers more time to haggle over whether to pass a straight increase in the ceiling or attach spending cuts as many Republicans want to do as part of it.

Russia investigations continue -- new FBI nominee to testify

The Senate confirmation hearing for FBI nominee Chris Wray is likely to draw much attention this week. The Senate judiciary committee is scheduled to consider his nomination on Wednesday, and the hearing will inevitably revive talk of Comey's firing by Trump . But it also comes after Trump talked to Russian President Vladimir Putin about his country's alleged interference in the US elections when they met last week.

The House and Senate Russia investigations get back in gear this week after taking the July 4 recess off. Members of the House intelligence committee, in particular, are expected to have a busy July as they continue their closed-door interviews with Democratic and Republican witnesses.

Also, the House will face pressure to take up a Russia sanctions bill that got stalled over a procedural issue in the days before Trump's bilateral meeting with Putin . Democrats accused Republicans of deliberately delaying final passage of the bill -- by citing an occasionally enforced rule known as "blue slip" -- until after that important first meeting between the two world leaders. Republicans denied that was their intent, insisting that the "blue slip" violation, which mandates that bills generating revenue begin in the House, had to be addressed first. An agreement was reached and the Senate passed a slightly tweaked version of the measure just before recess, and now Senate Democrats will push House Republicans to put the sanctions bill on the floor for a vote in the coming days.

Finally, the Senate armed services committee will hold a hearing Thursday examining allegations Moscow orchestrated a failed coup attempt in Montenegro last year. The ambassador to the US from that country, which is now part of NATO, is scheduled to testify.