JERUSALEM -- While there has been no official policy shift in Washington yet, many Israelis seem eager to make the most of President Trump’s pro-Israel statements.

The mere suggestion that the U.S. Embassy in Israel could move to Jerusalem has folks talking Tuesday morning at the mainly-Israeli “souk,” or market, in the holy city.

“It have to be here in Jerusalem -- every embassy -- not just of the U.S., every embassy,” Asaf Cohen told CBS News correspondent Seth Doane. “Because Jerusalem is the capital of the state of Israel.”

“This is the right thing to do -- this is the moral thing to do,” agreed Yaniv Mazor as he sipped a coffee.

Asked why he felt the move was a moral obligation, Mazor told CBS News it was “because you know that the Jewish people, okay, has a very long connection with the city of Jerusalem.

The city is home to some of the holiest sites for Jews, but also for Muslims and Christians, and it’s also claimed as a capital by the Palestinians.

Shifting the U.S. embassy to Jerusalem, then, would show the U.S. siding with Israel in one of the world’s longest-running conflicts, and the notion is so contentious it sparks disagreement even at Mazor’s coffee table.

“It’s a bad idea to move the move the embassy to Jerusalem,” Vered Ben-Artzi insists.

Why?

“Because we don’t need world war three.”

On a windy rooftop in Ramallah, in the Palestinian territory of the West Bank -- just six miles north of Jerusalem -- Doane met senior Palestinian advisor Husam Zomlot.

He worried the move could cripple the already-stalled Middle East peace process, and give extremists a new reason to lash out with violence.

“It further complicates our lives and it starts with President Trump on the very wrong foot,” Zomlot tells CBS News. “We wanted to start on the right foot. We wanted him to be a force for peace.”

Meanwhile, in Iraq, firebrand anti-U.S. cleric Muqtada al-Sadr, whose militiamen fought U.S. troops in the country with devastating effect during the occupation to topple Saddam Hussein, said Tuesday that moving the embassy to Jerusalem would amount to “a public and more-explicit-than-ever declaration of war against Islam.”

No country has its embassy in Jerusalem, as the status of the city remains a key point of disagreement in the stalled peace process.

Jerusalem Mayor Nir Barkat tells CBS News that his administration stands ready to help the U.S. with any move.

“Better late than never,” Barkat says. “Recognize Jerusalem as the capital of the Jewish people -- like America does in any other country it works with.”

Barkat has been in contact with the Trump Administration.

Asked whether he views Mr. Trump as a game-changer for his city and his country, Barkat says he thinks, “President Trump has the right vision -- aligning interest with Israel.”

But back at the souk, one of his constituents, teacher Melissa Fragiadaki, urges caution over making any moves too quickly.

“I do believe it will result with violence and anger, and it will slow down the peace process, even though I do believe that Jerusalem is the capital of Israel,” she says.