Also greeting Bush was Ehud Olmert, the Israeli prime minister, who said the bond between Israel and the US is "unshakable".
"Your policies have reflected a basic understanding of the challenges facing Israel in this troubled region and a solid commitment to our national security," he said.
In a boost for Israel, Bush said: "The alliance between our two nations helps guarantee Israel's security as a Jewish state".
The Palestinians object to recognition of Israel as a Jewish state, saying it rejects the rights of refugees to return to lost properties in what is now Israel.
Besides Israel, Bush will visit the Palestinian West Bank, Kuwait, Bahrain, United Arab Emirates, Saudi Arabia and Egypt after Egypt.
In the Gaza Strip, Hamas leaders described the visit as US endorsement to the Israeli occupation of the Palestinian territories.
The Hamas-controlled territory is noticeably absent from the tour, with the US considering Hamas a terrorist organisation and refusing to negotiate with its leaders.
Hamas is organising another anti-Bush rally on Wednesday, with a four-metre-long banner denouncing Bush as a "war criminal" serving as the centrepiece.
Bush, left, is hugged by Peres as Olmert, right,
described the US president as a mentor [AFP]
Fawzi Barhoum, a Hamas spokesman, told Al Jazeera: "All the time, the American administration provides all kinds of support to Israel by providing weapons and millions of dollars to support the Israeli occupation, the killings and arrests of the Palestinians as well as the destruction of the Palestinian democratic process.
"So our suffering and our tragedy here in Palestine emanate from these kinds of US support to the Israeli occupation."
Barhoum also called on Bush to respect the Palestinian democratic process and end Washington's interference.
In Gaza, thousands of Hamas supporters took to the streets on Wednesday to protest against Bush's visit, calling him the "butcher of the world".
Angry protesters burned American and Israeli flags and effigies of Bush and Olmert, including a portrait of Bush with the caption "War Criminal Go Home".
Speaking to Al Jazeera's David Chater in Ramallah on Wednesday, Khaleel Shaheen, a Palestinian political analyst, said: "For Palestinians the visit will not add anything new. The Americans have red lines that they cannot cross in their relations with Israel.
"This line permits Israel to continue its settlement activities and its aggression against the Palestinians in the West Bank and the Gaza Strip.
"The Palestinians do not expect much from this visit. People here are worried more about the roads that are going to be closed tomorrow rather than worrying about the content of this visit.
"This visit will give a push for both the Israeli prime minister and Palestinian president Mahmoud Abbas. Both leaders are both weak. We should know that Olmert is facing a hard time and needs this visit to strengthen his situation.
"Abbas also needs this visit in order to convince the Palestinians that there is something going on. At least there is contact with the Israelis.
"But we should notice that the Palestinians weakened their position by themselves. They chose to continue these negotiations, while Israel is continuing to build settlements.
"This is something comes in contradiction with the Palestinian stance and against the promise given to Palestinian people that there would be no negotiations as settlement activities going on."
The Israeli view
On the other hand, speaking to Al Jazeera's Jacky Rowland in west Jerusalem, Yossi Beilin, a Knesset member, said: "I do not think that President Bush is coming to solve any problem.
"In my view, I think what he is coming to do is to tell the world and tell us - the Palestinians and Israelis - that he cares, that he believes in the last year of his second term, there is a chance to make peace and to sign a peace treaty.
"This is of some importance after seven years in power, in which he did too little for the peace process in the Middle East.
Bush's challenge is to convince sceptical governments that he is willing to devote the time and effort necessary to bridge decades of differences in the region.
There has been little progress since Bush hosted a one-day meeting in November in Annapolis, Maryland, to launch the first major peace talks in seven years.
On the eve of Bush's arrival, Olmert and Mahmoud Abbas, the Palestinian president, pledged to have negotiators begin work immediately on the so-called final status issues.
These include the final borders between Israel and a future Palestine, completing claims to the city of Jerusalem, the fate of millions of Palestinian refugees and Israeli security concerns.
Source: Al Jazeera and agencies