This week! For three days only! President Obama plans to spend three days on Capitol Hill trying to make nice with lawmakers who've been feeling neglected and hoping that Republicans will accept that he's serious about cutting big deals on tax reform, spending cuts, gun control and immigration.

The president's high-profile lobbying effort begins with Senate Democrats on Tuesday, followed by a visit with House Republicans on Wednesday. Finally, Obama plans to visit the Hill for meetings with Senate Republicans and House Democrats on Thursday. (When he meets with Senate Republicans, he's expected to dine on Maine lobster, potato chips and blueberry pie, according to Sen. Susan Collins (R-Maine), who's responsible for the Thursday luncheon.)

After dining last week with selected Senate Republicans and House Budget Committee Chairman Paul Ryan (R-Wis.), are we entering a new era of good feelings? Not so fast.

“The proof will be in the coming weeks as to whether or not it is a real sincere outreach to find common ground,” Ryan told "Fox News Sunday."

On NBC's "Meet the Press," Rep. Cory Gardner (R-Colo.) put it in terms the kids could appreciate: “I don’t think we’re going to be doing the 'Harlem Shake' anytime soon together. I think that we can actually use this chance to see what’s going to happen.”

What else, if anything, is Congress doing this week? Here's a preview:

1. Budget Week: One reason why things may not go smoothly for Obama on Capitol Hill this week is because Ryan plans to unveil a budget plan that promotes repealing the 2010 health-care law. Ryan insisted Sunday on "Fox News Sunday" that Republicans remain serious about repealing the law -- despite the Supreme Court's decision to uphold the measure last summer.

The 2012 Republican vice-presidential candidate's budget plan is also expected to propose wiping out the deficit over the next decade, instead of 20 years, as he has proposed before. The proposal also calls for overhauling Medicare and Medicaid, as he suggested during last year's presidential campaign.

Senate Budget Committee Chairman Patty Murray (D-Wash.) plans to unveil her blueprint on Wednesday -- and it's not expected to do anything to erase the nation's deficits. As The Post's Lori Montgomery reported last week, Murray is expected to propose replacing the automatic budget cuts known as sequestration with higher taxes, drawn from overhauling the tax code under a fast-track process known as “reconciliation.”

So if Ryan is proposing an end to "Obamacare" and Murray wants to raise taxes while doing nothing to erase the deficit -- how exactly can we expect Congress to pass a budget?

2. Gun debate continues: On Tuesday, the Senate Judiciary Committee plans to continue debating the proposed ban on military-style assault weapons. Before adjourning last week, the panel approved a bipartisan bill to punish "straw purchasers," or people who knowingly buy firearms for someone legally barred from doing so.

In addition to the assault weapons ban (which is expected to ultimately fail in the full Senate), the Judiciary Committee plans to vote on a Democratic-sponsored plan to expand the nation's gun background check system and a proposal to expand a Justice Department program that provides federal dollars to school districts for security planning. Expect a marathon day of debate on Tuesday.

3. Will the Senate resolve to pass a continuing resolution?: The House easily passed its GOP-backed spending proposal last week, but Senate Democrats hope to make some changes. Senate Appropriations Chairwoman Barbara Mikulski (D-Md.) is expected to introduce her version of the "CR" on Monday, with a full Senate vote as early as Wednesday.

What are they key differences in the House and Senate plans? Here's how Rosalind S. Helderman expertly laid it out late last week:

For the most part, the House provided provisions to help blunt the impact of sequester largely only on the Defense Department. For most of the government, the House’s resolution would keep spending priorities that had been in place for the first half of the fiscal year — and then whack them by $85 billion, as President Obama ordered last week when the sequester took effect. Though the House restored no funding to the military, which absorbed half of the sequester reductions, it did embrace new priorities for military spending negotiated between Democrats and Republicans in the House and Senate late last year. That had the result of blunting the impact of the cuts for the Pentagon.

Mikulski said the Senate measure also would reflect new spending priorities in a few other areas–agriculture, commerce, justice and science, as well as homeland security. Plus, she said the Senate would a embrace new mechanism to give agency heads the ability to transfer money between programs, with Congressional approval. That would help agency heads manage the sequester, which lopped an equal percentage off every effected program, regardless of its importance.

But the Senate would give agencies no additional money and the funding bill would reflect the same $982 billion in funding, reflecting the sequester’s cuts, included in the House bill, she said.

Bottom line: The two sides must agree on a funding plan by Wednesday March 27, when the current measure keeping the government running expires. But Congress is scheduled to adjourn for a two-week recess on Friday March 22 -- making that date the deadline that congressional leaders have circled on their calendars.

For those of you less familiar with the term, "Continuing Resolution," check out this "Edsplainer" below:

4. Who's leaving, who's running in 2014?: As many expected, Sen. Carl Levin (D-Mich.) said last week that he won't run for reelection next year. The 78-year-old senator has held his seat since 1979 -- and hasn't faced a serious challenge in decades. Rep. Gary Peters (D-Mich.), the early Democratic favorite, said Friday that he'll seriously consider a bid to succeed Levin. Less certain is who might run on the Republican ticket. Several high-profile Republicans have already opted out, but one possibility, according to a thinly-sourced Roll Call report: Scott Romney, the brother of 2012 Republican presidential candidate Mitt Romney.

Ashley Judd, a University of Kentucky alum, kisses the UK mascot during a timeout at a basketball game in 2010. (James Crisp/AP)

Meanwhile in Kentucky, the buzz continues to focus on Ashley Judd, who is expected to announce plans to challenge Senate Minority Leader Mitch McConnell (R-Ky.) in the coming months, according to reports from over the weekend. If the actress jumps in, expect the Kentucky contest to become one of the most-watched, most-expensive and most-nasty campaign of the cycle.

(In case you missed it, make sure to read Aaron Blake's review of the recent wave of Senate retirements. Turns out that despite all those senior citizens retiring from the upper chamber, the average age of the Senate declined only by the slightest.)

What did we miss? What do YOU think will happen this week? Share your thoughts in the comments section below.

Follow Ed O'Keefe on Twitter: @edatpost