Leaders on both sides of the aisle have begrudgingly praised the Budget Control Act of 2011, and newspapers this morning hailed it as the deal to prevent default, but the bill isn’t quite “done and done.”

Supposedly, Speaker John Boehner hopes the House will vote first on the deal as a special “Message to the Senate,” possibly as early as 2 p.m., but that’s not confirmed yet, so stand by … In the meantime, let’s parse the chances of passage in both the House and Senate. (In case you missed the actual details of the fluid deal, here’s a solid summary.)

Vice President Joe Biden this morning headed to both chambers to confer with Democratic leadership to ensure the necessary votes would be there — a sign that the bill’s success isn’t quite a sure thing. And no wonder: Critics on both sides find the bill, at best, a disappointment and, at worst (from passage’s standpoint), utterly unpalatable.

But, for the first time since the House passed its original budget, the Senate doesn’t seem to be the biggest threat to passage of a deal, a reflection, perhaps, that this newest compromise was primarily brokered by the president and Senate Minority Leader Mitch McConnell (R-Ky.), with a relatively early sign-off from Senate Majority Leader Harry Reid (D-Nev.).

Sen. John McCain (R-Ariz.), for one, will be a “yes” vote on the bill he’s glad doesn’t require the passage of a balanced budget amendment (which he again insisted this morning on “Fox and Friends” is an impossible proposition as long as Republicans control just one-third of the government). Sure, he says he’ll have to “swallow hard” and he also claims he’s concerned about the defense spending cuts the bill could engender, but he’ll vote for it anyway. Other establishment Republicans will likely vote the same way — and the Democrats will line up behind Reid, as they always do.

But a few solutions-oriented senators will still vote against a bill that signifies a political accomplishment only. Sen. Mike Lee (R-Utah), for example, has said he’ll vote against it on the grounds that it does nothing to reform the way Washington spends taxpayer money. And Sen. Ron Johnson (R-Wis.) reportedly feels queasy at the thought of a vote on a “disgusting” debt ceiling increase of $2.4 trillion.

“Even right now we are supposedly talking about a deal that will be $2.4-trillion worth,” Johnson said last night on Fox News. ‘Think about what that is: $2.4-million-million worth. You just have a couple people negotiating this. I mean, it is ridiculous. We’re talking about the federal government’s budget here. … Ten years ago we spent $1.8 trillion. This year we will spend $3.6 trillion, and more. What the president’s budget would do after 10 years: Increase that from $3.7 to $5.7 trillion. He would spend $46 trillion. And all we are doing is lowering that by, what, $900 billion? … It may be a step in the right direction, but it is not fixing the problem.”

In the House, Democrats and Republicans alike aren’t amenable, either. Minority Leader Nancy Pelosi (D-Calif.), for example, met the most recent proposal with a cold reception, even though her former adviser Brendan Daly told me this morning he’s confident she’ll release her caucus to vote for it, even if she herself votes “no.”

On the right, Tea Party stalwarts like Rep. Joe Walsh (R-Ill.) and Rep. Louis Gomert (R-Tex.) have said they’ll vote against the bill. (“I’m looking for a reason to vote ‘yes’ and I can’t find it,” Gomert told Alisyn Camerota on “America’s Newsroom” this morning. He was quite literally “looking” — he stayed up past 2:00 a.m. studying the bill, he said.) The other 20 GOP representatives who voted against House Speaker John Boehner’s original compromise bill will also likely remain steadfast in the face of this new and still-more-watery legislation. More Republicans might defect.

The sticking point among slightly more “establishment” R’s in the House still seems to be the potential cuts to defense funding, which would go into effect should Congress fail to enact the recommendations of the compromise bill’s special commission to reduce the deficit. Importantly, House Armed Services Chairman Buck McKeon (R-Calif.) and other members of HASC are still weighing whether to vote “yes.”

But as AP pointed out last night, the Democrats who will support this Obama-McConnell last-minute deal will probably more than cover the loss of any additional Republicans. Plus, Boehner is fighting harder than any of the chief compromise crafters to sell this bill, so he’ll likely limit the number of Republican “no” votes.

We shall see …

Update I: Vice President Biden will speak in moments to report the results of his meetings with Democratic leadership earlier today. Still no word on when the actual vote will be. But, in the meantime, both Moody’s and Standard and Poor’s have suggested they’re less than happy with the deal as it is.

Update II: Looks like the House vote will be in the evening at the earliest … Via Guy Benson: “Joe Biden is meeting with House Democrats this afternoon to help sell the deal  in advance of a vote, which could come this evening.  Down the hall, Harry Reid says he’d also prefer to tee-up a Senate vote for today, but can’t commit to it.”

Update III: Nancy Pelosi is telling her people to “vote your conscience.” That means no real whipping from the vote-lassoing queen.

But Biden still sounds hopeful that the bill will pass. “I didn’t go to convince; I went to explain and lay out exactly how we got to where we were and why this is so important for the country,” he said moments ago, coming out of meeting with Democrats. “They expressed all their frustration, which I’d be frustrated if i were sitting there, as well, that it came down to the wire, as well. They asked about the proposed legislation. I thought it was a good meeting and I feel confident that this will pass.” Biden said the “one, overwhelming, redeeming feature” of the deal is that it prevents the debt ceiling from coming up for debate again until after 2012 — “and that has nothing to do with elections.”

Update IV: I just wrote a separate post about the House Republican leadership press conference, but the key takeaway is that Boehner’s sounding confident the deal will pass in the House.