A private session with President Barack Obama is a big diplomatic get — all the more so when it comes as world leaders are descending en masse on New York for the opening week of the United Nations General Assembly.

But two foreign leaders seem apprehensive, to say the least, about their meeting with Obama on Tuesday: Israeli Prime Minister Benjamin Netanyahu and Palestinian Authority President Mahmoud Abbas.

Netanyahu and Abbas did not dare decline a U.S. presidential invitation for a meeting with Obama, however conflicted the two leaders are about engaging in summitry before they feel conditions have been met for relaunching Middle East peace talks.

But both men also know that risks abound for them when they gather with Obama at the Waldorf Astoria hotel. With little or no chance for substantive breakthroughs at this largely cosmetic session, even the stagecraft could be dangerous. The iconic photograph that will doubtless emerge from the meeting, for starters, will be painstakingly scrutinized by highly skeptical domestic political audiences back home — audiences that are concerned that their leaders didn’t live up to demands for more concessions from the other side before agreeing to their first meeting, even one that does not commit them to anything.

For Obama, too, the trilateral meeting with the Israeli and Palestinian leaders is not without risks. Even a symbolic meeting and photo op raise expectations of what can be achieved.

U.S. officials have sought to lower expectations for the meeting, saying it is more of a symbolic than a substantive breakthrough and will not be accompanied by an announcement for the restart of Israeli-Palestinian peace talks. The hope is that the meeting leads to something positive, one U.S. official said on condition of anonymity Monday.

“Obama thinks he can extract some agreement outline and announce a follow-up,” a former Israeli diplomat said. “What seems certain is that the U.S. will not — at this stage — come out with a declaration of principles. The president could still make the Mideast a major segment of his speech either at the U.N. or even in Pittsburgh,” where Obama meets with G-20 leaders later in the week.

“Clearly, the Obama team wants to start the next chapter,” an American Jewish community leader told POLITICO on condition of anonymity. “I presume Bibi wants to make sure of two somewhat contradictory things: that he has demonstrated to Abbas and all Arabs that, actually, Obama — even Obama — cannot ‘deliver’ Israel; and that he is not to ‘blame’ for any kind of failure to reach a deal or relaunch of talks.”

“On the other hand, Abbas wants to keep the pressure on Bibi — he is to be blamed for no breakthrough and he is to be blamed for no deal on settlements. ... The best headline the White House can hope for is something like: ‘Obama Meets With Netanyahu and Abbas; They Agree on Continued Efforts Toward Restarting Peace Talks.’”

Netanyahu will have a chance to spin his take on the three-way conference when he meets with Jewish community leaders Thursday afternoon at the 92nd Street Y.