For Mr. Romney, the trip is a critical opportunity to show statesmanship, especially after a less-than-perfect London trip, and to highlight his relationship with Prime Minister Benjamin Netanyahu of Israel, whom he frequently describes as an “old friend.” But the perils include appearing critical of a sitting president from foreign soil.

For Mr. Netanyahu, a right-leaning leader whose relationship with President Obama has been rocky at best, the visit is a chance to ratchet up the pressure on the administration over the Iranian nuclear threat, but he must be careful not to be seen as partisan or meddling.

The diplomatic dance will have two dips: a work meeting at the prime minister’s office on Sunday morning, and a meal that evening with their wives. Analysts will be monitoring not only what is said — and what is not said — but every wink, nod and backslap that could feed the existing narrative of a bond between the two men.

“The choreography of a thing like this is complicated,” said Robert Shrum, a veteran Democratic political consultant in the United States who has also run campaigns in Israel. “You’ve got to look like you know what you’re talking about, you’ve got to convey your positive message, and you’ve got to avoid all the land mines — there are Israeli domestic land mines, and there are land mines back here.”

Summer pilgrimages to Jerusalem by aspiring commanders in chief are standard fare — Mr. Obama came here almost exactly four years ago — but the inclusion of a fund-raiser is more unusual. Organizers say there will be 20 to 30 people at the $50,000-per-couple event at the posh King David Hotel, including American citizens who live part time or full time here, and some American Jews, led by the billionaire Sheldon Adelson, who are flying over for the visit.