Story Highlights• House Speaker Nancy Pelosi visits market, sacred mosque in Damascus
• Syrian media hail Pelosi's visit as important; Bush calls it "counterproductive"
• Pelosi is set to meet with Syrian President Bashar Al-Assad on Wednesday
• Democrat says trip was intended to build confidence between U.S. and Syria
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DAMASCUS, Syria (CNN) -- House Speaker Nancy Pelosi arrived Tuesday in the Syrian capital, where she is expected to tackle an issue complicating the U.S. mission in the Middle East -- Damascus' alleged support for militants in Iraq and Lebanon.
Though her visit conjured a strong rebuke by the White House, Syria seemed to welcome the 67-year-old California Democrat.
After arriving in Damascus, Pelosi visited a centuries-old market district, where she mingled with Syrians. She also visited the Omayyad mosque in Old Damascus, which is considered one of the most sacred sites in the capital. (Watch what the White House said last week about Pelosi's trip )
The Syrian media are hailing her visit as a potential breakthrough in icy U.S.-Syrian relations, with the Syria Times calling her a "brave lady on an invaluable mission."
Even the state-run news agency harped on the importance of Pelosi's visit. Headlines from the Syrian Arab News Agency read: "World and Arab newspapers stress importance of Pelosi's visit to Syria" and "Pelosi describes her visit to Damascus as important."
The House speaker is scheduled to meet with President Bashar Al-Assad on Wednesday, making her the highest-ranking American politician to meet with a Syrian leader in more than 12 years. In 1994, President Clinton sat down with Al-Assad's father, the late Syrian President Hafez Al-Assad.
President Bush admonished Pelosi in a Rose Garden question-and-answer session Tuesday, calling her trip "counterproductive."
"Going to Syria sends mixed signals, signals in the region and, of course, mixed signals to President Assad," Bush said. "And by that I mean, you know, photo opportunities and/or meetings with President Assad lead the Assad government to believe they're part of the mainstream of the international community, when in fact, they're a state sponsor of terror."
He further accused Syria of destabilizing the Lebanese government and turning a blind eye to foreign fighters flowing across its border into Iraq.
American and European officials have met with Al-Assad in the past, "and yet we haven't seen any action," Bush said. "The best way to meet with a leader like Assad or people from Syria is in the larger context of trying to get the global community to help change his behavior."
Bush repeatedly has rejected the prospect of talking with Syria despite a December report by the Iraq Study Group that recommended multilateral diplomacy with Iraq's neighbors, including Iran and Syria, as a means of improving stability in the war-torn nation.
During a stop in Beirut, Lebanon, on Monday, Pelosi said her trip is intended as a bid to build confidence between Washington and Damascus. Pelosi stood by the U.S. assertion that Syria supports groups that the United States considers terrorist organizations.
"Of course the role of Syria in Iraq, the role of Syria supporting Hamas and Hezbollah, the role of Syria in so many respects -- we think there could be a vast improvement," she said. "We think it's a good idea to establish the facts, to hopefully build some confidence between us. We have no illusions, but we have great hope."
Syria, which since 1979 has been on the U.S. State Department's list of nations that sponsor terror, acknowledges that it provides funding for the Palestinian and Lebanese militant organizations, but it denies providing either group with arms.
Another topic on Pelosi's agenda will be the plight of Israeli soldiers captured by Lebanon's Hezbollah in July. The incident sparked a war with Israel, and both Washington and Israel accused Syria of arming and funding Hezbollah during the conflict. (Full story)
Syria was a major military and political force in Lebanon for almost 30 years until after Lebanese Prime Minister Rafik al-Hariri's assassination in 2005. A wave of protests called the "Cedar Revolution" ensued, and Syria was forced to pull its roughly 15,000 troops out of Lebanon.
U.N. investigators linked al-Hariri's killing to Syria and its Lebanese allies, but Damascus has denied any role in the slaying, which it says it condemns.
Pelosi and her delegation visited officials in Israel on Sunday, Lebanon on Monday and the West Bank on Tuesday, according to media reports.
The delegation includes one Republican, David Hobson of Ohio. Other Democratic members include Reps. Henry Waxman and Tom Lantos of California, Louise Slaughter of New York, Nick Rahall of West Virginia and Keith Ellison of Minnesota, the first Muslim to serve in Congress.
CNN's Brent Sadler contributed to this report.