Al Jazeera's Clayton Swisher reported that Carter's talks with Meshaal ran over for more than an hour and a half without a break in an atmosphere described as "pleasant".
"Al Jazeera has learned, based on exclusive sources inside the Carter-Hamas talks, that a delegation of Hamas officials from the Gaza Strip, led by the Hamas foreign minister Mahmoud Zahar, is going to arrive in Damascus on Saturday to join the talks with President Carter and Hamas leadership," Swisher said.
He added that the talks covered four key areas, which are rocket attacks inside Israel, exchange of prisoners, lifting the economic siege against the people of Gaza as well as a proposal by an Israeli minister to meet Hamas to discuss the fate of an Israeli soldier held by the Palestinians.
Eli Yishai, the Israeli deputy prime minister, said he was ready to meet Meshaal to negotiate the release of prisoners held by the movement, according to Friday's Haaretz daily newspaper.
The newspaper quoted Yishai as telling Carter: "I am ready to meet with all necessary Hamas members."
US reactionier on Friday, Hamas said Gilad Shalit, the Israeli soldier captured by fighters in the Gaza Strip, would "not see the light" until Palestinian prisoners were released in a prisoner exchange.
The US administration played down the importance of the Carter-Meshaal meeting.
David Foley, the US State Department spokesman for Near East Affairs, told Al Jazeera: "We've been very clear about our position about Jimmy Carter. Former President Carter is a private citizen and is on a private visit. He does not have the support of the United States government."
On the possibility of considering Hamas as peace partner, Foley said: "We want to see Hamas part of the peace process, to be a partner for peace with the Israelis that will lead to two states side by side living in peace and security.
"We have set out together with the European Union, the Russians and the United Nations very simple conditions for Hamas to begin to address the very difficult challenges and frankly the compromises that are going to be necessary.
"We need to see Hamas simply do a couple of things first and that will show that they are actually serious about peace."
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.
Carter had met a delegation of Hamas leaders in Cairo on Thursday, as well as holding talks with Husni Mubarak, the Egyptian president.
Carter defended his meetings, calling them necessary in the Israeli-Palestinian peace process and described Israel's siege of the Gaza Strip as a crime.
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, democratically elected into power in January 2006, have been counterproductive.
His meeting with Hamas 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".
The US welcomed the Russian engagement in the peace process.
Foley told Al Jazeera: "Russia is a very valued member of the Quartet. They've been part of these [peace] discussions. Certainly we will welcome Russian engagement."
Israel has sealed off the West Bank and Gaza for 10 days as it celebrates the Jewish holiday of Passover.
Tanks and soldiers moved into position at the crossings as the shutdown took effect from early on Friday, barring Palestinians in 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 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, but human rights groups argue that the siege of Gaza is collective punishment of the populace.