The Hamas rulers of Gaza and the rival Palestinian Authority leadership of the West Bank rarely see eye-to-eye on anything. But with mass protests rocking Egypt, across Gaza’s southern border, the Palestinian adversaries have united in maintaining a cautious silence, hedging their bets given the unpredictability of the outcome and clearly concerned about a possible spread of popular unrest to their areas.

Both Hamas and the Western-backed Palestinian Authority have prevented popular demonstrations in support of protesting Egyptians in recent days, apparently preferring to show a front of Palestinian neutrality and worried that things could spin out of control.

In an early sign that people on both sides were seeking to capitalize on the regional turmoil, Palestinians inspired by how social networking sites helped to mobilize demonstrators in Egypt and Tunisia created two pages on Facebook over the weekend, one urging people to rebel against the Islamic militant rulers of Gaza, and the other against the Palestinian Authority in the West Bank.

By mid-Monday, several thousand fans had registered for the anti-Hamas page, Preparations for Al-Karama (Dignity) Revolution in Gaza, which called for mass protests in Gaza after Friday prayers on Feb. 11. The anti-authority page, Preparations for Revolution against the Zionist-Fatah Authority, called for protests after prayers in the West Bank this Friday, and had attracted a few hundred fans.

Apparently nervous, the Hamas police dispersed a handful of demonstrators who gathered in Gaza city on Monday afternoon to show support for the Egyptian people. The bearded plainclothes officers called in a group of female officers and arrested three young female demonstrators, a human rights advocate and another male demonstrator. The call for that demonstration was also made through Facebook.

The schism between the Palestinian sides deepened after Hamas won parliamentary elections in 2006 then seized full control of Gaza a year later, routing pro-Fatah forces after months of factional fighting.

Hosni Mubarak, the embattled president of Egypt, has been a strong ally of the Palestinian Authority and a staunch supporter of the now-stalled Israeli-Palestinian peace process. Mahmoud Abbas, the Palestinian president, called Mr. Mubarak on Saturday to express his hope that calm and stability would be restored. Egypt has also brokered reconciliation talks, so far without success, between Hamas and Fatah, the party led by Mr. Abbas.

Palestinian leaders on both sides have been wary of speaking publicly about Egypt, aware of what some local observers were calling the Kuwait effect. They were referring to the mistake made by Yasir Arafat, then the Palestinian leader, who sided with Iraq when Saddam Hussein invaded Kuwait in 1990. Kuwait withdrew its support for the Palestinians and expelled hundreds of thousands who were living and working there.

Mahmoud Zahar, a Hamas leader in Gaza known for his often fiery outspokenness, said on Monday, “We do not intervene in Egypt’s internal affairs.”

Nabil Shaath, a member of the Fatah Central Committee and Fatah’s foreign relations commissioner, said in a telephone interview from the West Bank city of Ramallah on Sunday: “We are praying that Egypt comes out of this in the final analysis with more freedom, security, democracy, unity. It is their decision, not ours. Our position has been very circumspect — we are very much aware of the difficulties Egypt is going through and we cannot take sides.”

“For us,” he added, “Egypt is an existential relationship like that between Israel and the United States.”

Many analysts have noted that if the Muslim Brotherhood eventually comes to power in Egypt, Hamas would be greatly strengthened, and could take over the West Bank. Mr. Shaath said that talk of a Muslim Brotherhood-run Egypt was “exaggerated.” He also denied any concern that anti-establishment protests could spread to the West Bank, saying that the Palestinians had enough problems with the Israeli occupation and the West Bank-Gaza divide.

Still, Mr. Abbas met with his security chiefs on Sunday. And in what appeared to be an effort to head off criticism of the veteran leadership in the West Bank, Fatah officials for the first time in months began talking about the possibility of holding long-overdue elections.

Azzam al-Ahmed, a senior Fatah official, told the Voice of Palestine radio on Monday that the leadership was discussing the possibility of holding presidential and parliamentary elections, despite the internal division. Mr. Abbas was elected in 2005; he said in late 2009 that he would not seek re-election as president, and scheduled elections for January 2010. But when it became clear that Hamas would not cooperate with staging the elections in Gaza Mr. Abbas’s term was extended indefinitely.

The authority leaders faced a challenge of their own this month after the satellite channel Al Jazeera published leaked documents showing concessions made by Mr. Abbas’s negotiators in past negotiations with the Israelis.

The Palestinian Authority allowed one large demonstration to take place Friday in the West Bank city of Hebron, against Al Jazeera and in support of Mr. Abbas. In a display of force dozens of armed Palestinian Authority security force members rode through the streets of Hebron, long considered a stronghold of Hamas.

The official Palestinian Authority TV has avoided showing the scale of the protests in Egypt. Emad al-Asfar, the director general of programming, said the station did not want “to interfere in the internal affairs of governments” and was trying to avoid “inciting the public through live coverage” and “unauthenticated stories.”

Nevertheless, Palestinians in the West Bank were mostly supportive of the Egyptians seeking change.

“We are happy,” said Rashad Zaid, 20, a university student from Ramallah, “because the barriers of fear have collapsed and people are able to raise their voices against those who have acted brutally. We hope that the street movement achieves its goals.”

Other Palestinians expressed concern that an end of Mr. Mubarak’s rule would leave Egypt in chaos.

Many Israeli analysts say that the regional turmoil precludes peace moves and territorial concessions in the near future, putting off prospects of a deal with the Palestinians. But few Palestinians seemed concerned about that, noting that negotiations have been suspended for months in any case.

In Gaza, where 1.5 million Palestinians rely heavily on the border with Egypt for transit and certain supplies, there were fears of increasing prices and of being stuck, as some smuggling tunnels under the border and the official above-ground border crossing were shut.

Hamas deployed forces along Gaza’s border with Egypt, preventing tunnel operators from reaching the area.

But five Palestinians who fled from Egyptian prisons had arrived back in Gaza by Monday, entering via the tunnels.

One was Mohammed Al-Shaer, known as the “lord of the tunnels,” a veteran smuggler who was arrested in Cairo in September 2009.