By James Mackenzie

JERUSALEM (Reuters) - Israel's right-wing government has maintained a diplomatic silence this week as U.S. President Donald Trump fired off a blizzard of announcements that have shaken Israeli assumptions about their country's standing with its most important ally.

Trump's decision to bypass Israel during his current visit to the Middle East had already been seen as a marker of the his administration's increased focus on lucrative business deals with wealthy Gulf countries, including Qatar, which Israeli officials have long accused of helping Hamas.

Even before the trip began, Israel was on edge over U.S. talks with its arch-enemy Iran and over Trump's decision to stop bombing the Houthis in Yemen, regardless of the Iranian-backed group's determination to keep up its own missile strikes against Israel.

Israeli officials were then forced to stand by and watch as the United States negotiated to reach a deal with the Palestinian militant group Hamas to bring home Edan Alexander, the last surviving American hostage in Gaza.

Since then, they have had to listen as Trump declared an end to sanctions on Syria and called for a normalization of relations with the new government in Damascus, which Israel regards as a barely disguised jihadist regime.

Even as Trump spoke in Riyadh on Tuesday, claiming credit for the ceasefire agreement with the Houthis, Israeli media noted that warning sirens were sounding in areas across Israel including Jerusalem and Tel Aviv as a missile from Yemen headed towards them.

Trump himself has brushed off any suggestion of a break with Israel, telling reporters accompanying him in the Gulf that his visit would ultimately benefit a country that has so far viewed him as one of its staunchest supporters.

"This is good for Israel, having a relationship like I have with these countries; Middle Eastern countries, essentially all of them," he said.

Israeli Prime Minister Benjamin Netanyahu has so far made no comment, beyond thanking Trump for helping with the release of Alexander. 

But he has faced widespread public perception that Israel, already under international pressure over the Gaza war, which has stymied its own hopes of normalizing relations with Saudi Arabia, has been left behind.

"The Middle East is in the process of being reshaped in front of our very eyes through a series of agreements and meetings, while Israel has remained (in the best-case scenario) an observer on the sidelines," wrote Yoav Limor, a commentator in the right-leaning Israel Hayom newspaper.

DIVERGING PRIORITIES

Netanyahu, currently on trial on corruption charges which he denies, made no secret of his preference for Trump over the previous White House incumbent Joe Biden, who held back some heavy munitions deliveries and imposed sanctions on a number of violent Israeli settlers.

He faces pressure both from religious-nationalist hardliners in his government, who have insisted on continuing the war in Gaza until the final defeat of Hamas, and an Israeli public increasingly tired of a conflict that has lasted for more than 18 months. So far, he has sided with the hardliners.

But the events of the past two weeks suggest there was a "clear divergence of priorities" and the special treatment from U.S. administrations that has normally been enjoyed by Israel may not apply, said Jonathan Panikoff, former deputy U.S. national intelligence officer for the Middle East.

"Trump is clearly determined to move ahead with a transactional, trade and investment focused agenda," said Panikoff, now at the Atlantic Council think-tank in Washington.

"If the traditional political or security matters that the U.S. and Israel have always historically coordinated on closely don't align well with Trump's priorities, he's going to go forward with them anyways."

While insisting that U.S.-Israeli relations remain strong, Trump administration officials at times have privately expressed frustration with Netanyahu as the president seeks to fulfil his campaign promise to quickly end the wars in Gaza and Ukraine.

They want Netanyahu to work harder to reach a ceasefire and hostages deal with Hamas and have also shown little appetite for backing any Israeli strike on Iran's nuclear facilities while the U.S. pursues a diplomatic solution.

White House National Security Council spokesman James Hewitt said the administration was continuing to work with Israel to free the remaining 58 hostages held in Gaza and to strengthen regional security in the Middle East.

"Israel has had no better friend in its history than President Trump," he said.

The hardliners in the Israeli government, who once rejoiced at Trump's announcement of a plan to clear Gaza of its Palestinian population and develop the coastal enclave into a beachside resort, have been largely silent and Israeli officials have been careful to avoid any criticism of the U.S. administration.

"The United States is a sovereign country," a foreign ministry spokesperson said this week, when asked if there was concern that Israel had been sidelined over the release of Alexander. Israel's "intimate dialogue" with the United States would be conducted "directly and not through the media."

An Israeli team has been dispatched to Doha to join ceasefire talks coordinated by Trump's special envoy Steve Witkoff, but Israeli forces have stepped up strikes in Gaza, killing dozens of Palestinians on Wednesday.

Netanyahu himself signalled Israel, which earlier this month announced plans for an intensified campaign in Gaza, was sticking to its war aims, including dismantling Hamas as a military and governing power.

"Israel will not stop and will not surrender," he said on Wednesday.

(Additional reporting by Matt Spetalnick in washington; Editing by Toby Chopra)