http://news.bbc.co.uk/2/hi/middle_east/7354027.stmHamas spokesmen said Mr Carter had asked for it to stop rocket attacks on Israel and to enter talks for the release of an Israeli captive.
They said any truce must be two-way and there would be a "price" for freeing Israeli soldier Gilad Shalit.
Mr Carter earlier met Syrian President Bashar al-Assad
But leading Hamas political bureau figure, Mohammed Nazzal, told reporters: "Carter suggested a truce and that Hamas should stop its rockets against Israel.
"We support a truce, but Israel should support it too."
One senior Hamas official in Damascus told Associated Press news agency Mr Carter had also asked Hamas to agree to a meeting with Israeli Deputy Prime Minister Eli Yishai to discuss a prisoner exchange for captured Israeli soldier Gilad Shalit.
Mr Nazzal would only say that Hamas leaders were to meet later to discuss the fate of Cpl Shalit, who was captured by Hamas in a raid into Israel from Gaza two years ago.
"They will discuss details related to the price and mechanisms for his release, which will not happen without a price," Mr Nazzal said.
From the Guardian
But in a proposal passed to Carter this week, an Israeli cabinet minister offered to meet high-level Hamas officials to ask for the release of a soldier being held by Hamas, a move which would depart from official Israeli policy. Carter, who won the Nobel Peace Prize in 2002, raised the issue of Corporal Gilad Shalit with Meshaal.
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