Liberals hope Kennedy -- who votes with them on many hot button social issues -- will stay on.
Conservatives want the 80-year-old centrist to step down and give President Donald Trump a chance to reshape the court in a more conservative mold.
"Justice Kennedy is the most powerful jurist in the world," said law professor Jessica Levinson of Loyola Law School. "The longer he stays on the court the greater the chance he can shape the legal doctrine for years to come."
But Kennedy didn't say anything in public Monday, and the court's actions might provide good reasons for him to stay.
Take the travel ban for instance. Kennedy voted to allow part of it to go in effect as it applies to foreign nationals with no "bona fide" connection to the United States. But three of his colleagues -- who wanted the entire ban to go into effect -- think that the "compromise" that Kennedy joined will cause chaos as parties in the coming days and weeks grapple with what actually constitutes a "bona fide" connection. Kennedy might see the need to help shepherd in a resolution to the complicated case concerning the role of the executive when the court hears the full case this fall.
"The world -- and justice Kennedy -- likely suspect that both those challenging and defending the so-called travel ban are arguing to an audience of one," Levinson said.
The court also agreed to take up a case that touches on Kennedy's landmark 2015 opinion clearing the way for same-sex marriage.
The case concerns a baker who objected to creating a specialty cake to celebrate the wedding anniversary of a same-sex couple.
On top of that, the court will hear a partisan gerrymandering case.
While the court has set a standard for unlawful racial gerrymanders, it has never come up with a test that goes to how much politics can play a role in the drawing of district lines after each census.
It's an issue Kennedy has been watching for years.
"He was the lone voice at the center holding the courthouse doors open in 2004, and he was the one who specifically asked lawyers to come back with more cases using different standards," said election law expert Justin Levitt, also of Loyola Law School.
Every term offers what one former Supreme Court nominee called an "intellectual feast" of cases, which may be a reason it's so difficult to leave the court.
Those who want Kennedy to stay on hope the next term will lure him back to the bench for at least another year.
Who would replace?
Talk of Kennedy's possible retirement has the parlor game of who would Trump nominate going at full tilt.
White House press secretary Sean Spicer said the list of Supreme Court nominees the Trump campaign released last May and September would play a "strong part" in filling future openings, but wouldn't promise that no new additions will be made to that list if a vacancy becomes available.
Though Spicer said the list "proved to be a very helpful" to the President the first time around, he also noted that he "can't say that there won't be someone added on or not" during an off-camera press briefing on Monday.
That contradicts what Trump told The Washington Times on April 30, which was that he would stick with the same list of nominees if another vacancy became available on the Supreme Court.
"Yes," Trump said. "That list was a big thing."
"It's a great list. From the moment I put that list out, it solved that problem. And I was proud to say it was my idea."