Stevens, who turns 90 Tuesday, told the president April 9 he was stepping down from the high court when the justices begin their summer recess in late June.
Obama's chance to nominate a second Supreme Court justice sets the stage for some real political bloodletting on the Senate floor. When a president is weak -- and with his approval ratings hovering around 50 percent Obama could be described as semi-weak -- individual senators, even senators in the president's own party, can turn combative.
In what seems to be an attempt to head off some of the rancor, the White House says the president "has invited the bipartisan leaders of the Senate and the bipartisan leaders of the Senate Judiciary Committee on Wednesday ... to discuss the Supreme Court vacancy left by the retirement of Justice Stevens."
The statement says Senate Majority Leader Harry Reid, D-Nev., Sen. Minority Leader Mitch McConnell, R-Tenn., Senate Judiciary Chairman Patrick Leahy, D-Vt., and Sen. Jeff Sessions, R-Ala., ranking minority member on the judiciary committee, "will travel to the White House on Wednesday morning for the meeting."
However, previous attempts to bring Republicans and Democrats on board for a particular Obama initiative, such as the White House summit on healthcare legislation, have been less than successful.
Some Republicans, including Sessions, are refusing to rule out a filibuster of a Supreme Court nominee.
No Supreme Court nominee has been filibustered in the Senate since Justice Abe Fortas was nominated to succeed Chief Justice Earl Warren in 1968. Fortas was believed to be too close to President Lyndon Johnson, and was accused of accepting speaking fees provided by companies who might have business before the high court.
Republicans and conservative Southern Democrats joined the filibuster, and when Democratic leaders failed to muster the votes needed for cloture, or an end to the filibuster, Fortas asked that his nomination be withdrawn. Fortas resigned from the court the next year in the face of new scandal.
In the modern era, 60 Senate votes are needed for cloture. The Democrats can count 59 votes in the Senate, but that includes Sen. Joe Lieberman, Ind-Conn., who caucuses -- sometimes -- with Democrats, and even Democrats are shaky on some key votes.
Other Supreme Court nominees have been voted down, most famously G. Harrold Carswell and Clement Haynsworth, both nominated by President Richard Nixon. Other nominations were withdrawn after negative reactions, particularly White House counsel Harriet Miers during President George W. Bush's administration, who was criticized from the left and the right.
In the current process, Sessions ominously has refused to rule out a filibuster for any Obama nominee.
Asked on "Meet the Press" whether there would be a fight over Obama's Supreme Court pick, Sessions said: "You know, the answer to that is in the president's hands. I think Senator Feinstein said it well the other day. She said that she believed the president should nominate somebody that would get a very strong vote, 70-plus votes, bipartisan ... But if we have a nominee that evidences a philosophy of 'judges know best,' that they can amend the Constitution by saying it has evolved, and effectuate agendas, then we're going to have a big fight about that because the American people don't want that."
Asked whether he would rule out a filibuster, Sessions said: "I hope we do not do that. I voted against (successful Obama nominee Justice Sonia) Sotomayor, but that was not a filibuster, and I think we'll just see how it plays out. Depends on the quality of the nominee."
In comments on an eventual nominee, Obama said, "It will also be someone who, like Justice Stevens, knows that in a democracy powerful interests must not be allowed to drown out the voices of ordinary citizens."
Sessions told "Meet the Press," "I'm not sure exactly what (the president) meant by that."
If a filibuster plays out, Sessions might have allies supporting him, such as Sens. Tom Coburn and James Inhofe, both R-Okla., and the combative Sen. Jim DeMint, R-S.C., who told fellow Republicans if they could stop healthcare legislation it would "break" the president.
Sen. Jon Kyl, R-Ariz. and the Senate minority whip, also refuses to rule out a filibuster.
Appearing Sunday on ABC's "This Week," Kyle said: "I'm never going to take it (a filibuster) off the table because of what the Democrats have achieved here, which is the possibility of a filibuster. President Obama himself attempted to filibuster Justice (Samuel) Alito, who now sits on the Supreme Court. So if the president isn't going to take it off the table, I'm not going to take it off the table. But I think it can easily be avoided by appointing, frankly ... (by nominating) someone who is mainstream enough that with intellect and the application of good law can persuade colleagues to support his position or her position."
On the same program as Sessions, Leahy agreed a nominee should unite, not divide, the country.
"If Republicans and Democrats want to set aside politics, stop listening to the single-issue groups of either the far right or the far left, they can," he said.
House Minority Leader John Boehner, R-Ohio, has not commented on the impending fight yet, but can be a tough infighter. Last month, he called Democratic employees on Capitol Hill "little punk staffers."
One calming voice in any potential dust-up is Sen. Orrin Hatch, the quietly reasonable Mormon Republican from Utah. Hatch told the "Today Show" he had "heard the name Hillary Clinton" as a possible Supreme Court nominee.
"That would be an interesting person in the mix," he said.
Clinton has been mentioned in a number of reports as a possible Stevens replacement -- though why she would trade the high-profile position of U.S. secretary of state for the lifetime interment and relative anonymity of the Supreme Court is anybody's guess.
The White House ruled her out as a potential high court nominee last week.
Hatch has been making peaceable noises about being fair to any Obama pick, but though he hasn't participated in any filibuster of a Supreme Court nominee he has blocked nominees for the lower federal courts.
U.S. Solicitor General Elena Kagan is believed to be on Obama's short list of possible Supreme Court nominees.
The U.S. Court of Appeals for the D.C. Circuit is considered the second most important court in the United States. In June 1999, President Bill Clinton nominated Kagan to an open seat on the D.C. Circuit, but then-Senate Judiciary Chairman Hatch refused to let the nomination come to the floor for a vote.
Hatch had no apparent personal animosity for Kagan. He just wanted to keep the seat open for a future Republican president to fill.
The seat stayed open until President George W. Bush filled it with John Roberts Jr. in 2003, then elevated Roberts to be chief justice on the Supreme Court in 2005.
Meanwhile, ABC News reports the White House says it will name a nominee by early next month.
Quoting sources, ABC, The Washington Post and a number of other media said fewer than 10 nominees, six of them women, are on Obama's short list.
A broad array of news outlets is predicting a hard fight in the Senate no matter whom the president picks. Obama appears to be boxed in to a struggle, no matter which way he jumps.
If he selects a liberal candidate who reflects his own political philosophy, he runs afoul of senators like Sessions who say such a nominee would be bad for the country. If he selects a moderate or a moderate conservative, he will be accused by liberal elements in his own party of dragging the Supreme Court to the right.