A senior Hamas official said that "the meeting between Carter and Meshaal will mainly focus on the Palestinian cause and means to deal with the Israeli occupation and its sanctions on the Palestinian people".
Mohammed Nasr said the two men would also discuss Gilad Shalit, an Israeli soldier captured by Palestinian groups in June 2006.
Nasr said: "In my opinion, the two are expected to discuss the Israeli soldier Shalit. I think it is natural to talk about this."
Earlier on Friday, Hamas said Shalit would "not see the light" until Palestinian prisoners were released in a prisoner exchange.
Mushir al-Masri, a Hamas official, said in a speech in Gaza: "Gilad will not see the light, will not see his mother, will not see his father, God willing, as long as our heroic prisoners do not see their families, in their houses."
Egyptian-led talks over a prisoner swap have been bogged down. Hamas has demanded the release of hundreds of prisoners. Israel has agreed to release some inmates, but has balked at some of those on Hamas's list.
On Thursday, Carter met a delegation of Hamas leaders from the Gaza Strip in Cairo and held talks with Husni Mubarak, the Egyptian president.
Speaking to the media, Carter defended his meetings, calling them necessary in the Israeli-Palestinian peace process.
The former US president also described Israel's siege of the Gaza Strip as a crime and an atrocity.
He said Palestinians in Gaza were being "starved to death", receiving fewer calories a day than people in the poorest parts of Africa.
"It's an atrocity what is being perpetrated as punishment on the people in Gaza," he said. "It's a crime... I think it is an abomination that this continues to go on."
Carter said that US attempts to undermine Hamas have been counterproductive.
His meeting with leaders of Hamas, a group he insists must be included in peace negotiations, has angered many in Washington and Israel, who call the Palestinian group a "terrorist" organisation.
In Moscow, Vladimir Putin, the Russian president, met Mahmoud Abbas, the Palestinian president, to discuss plans for a Middle East peace conference, as Moscow seeks a greater diplomatic role in the region.
Abbas, left, wants Russia to play a greater role
in the Middle East peace process [Reuters]
A Kremlin official told journalists on condition of anonymity that "special attention will be paid to. .. possible steps by Russia, including its initiative to hold a Moscow meeting on the Middle East".
The official said Putin and Abbas would discuss how to stabilise the situation and restore Palestinian unity.
Earlier in his visit, Abbas said a Moscow conference was urgently needed as Israeli-Palestinian talks started in Annapolis in the United States last November were "not advancing at the required pace or yielding the necessary progress".
Separately Abbas told the newspaper Nezavisimaya Gazeta that the conference would "be evidence of Russia's significance in the region and would strengthen her role in the peace process".
Israel has meanwhile sealed off the West Bank and Gaza for 10 days as it prepares to celebrate the Jewish holiday of Passover.
Tanks and soldiers moved into position at the crossings following the shutdown that took effect early on Friday, barring Palestinians from the occupied territories from entering Israel.
The shutdown follows a statement by Hamas on Thursday that "all options are open" to avenge the killing of 20 Palestinians in a day of air assaults and ground battles in Gaza that also left three Israeli soldiers dead.
In a statement published on the internet, the armed wing of Hamas called on its fighters to attack Israel "in every place and with all means available".
"This enemy only understands the language of force," it said.
The sealing of the borders also comes after the Israeli military said on Thursday that it had foiled a third attempt in the past week by Palestinian fighters to infiltrate the Gaza crossing.
The army said three Palestinian fighters attacked Kerem Shalom, a crossing used to deliver humanitarian supplies to Gaza.
One attacker was killed while a second was wounded when Israeli troops opened fire. The third man escaped, the military said.
The Israeli military said troops would be on high alert during the holiday period, regarding it "as a highly sensitive time, security wise".
It said it would try its best to preserve the daily life of Palestinians, allowing humanitarian cases, doctors and lawyers to cross the border.
But Israel has already been restricting the flow of food and fuel supplies into Gaza since Hamas seized control of the area last June.
Israel says the restrictions are in response to repeated rocket attacks launched from there and they have been tightened in recent weeks in response to continued heavy fighting.