She added: “They have begun to grapple with the core issues that can only be resolved through face-to-face negotiations.
"I believe they are serious about reaching an agreement that results in two states living side by side in peace and security."
Yesterday’s talks in the Egyptian resort of Sharm el-Sheikh closed without agreement as to whether Israel’s construction freeze in West Bank settlements would be extended.
But Mrs Clinton, who met Israeli President Shimon Peres this morning, said she believed peace was possible.
"We are convinced that the legitimate aspirations of these two peoples are not incompatible.”
Her comments came as Hamas claimed responsibility for firing two mortar shells from the Gaza Strip into southern Israel early this morning. Two rockets were also fired into Israel, one of which landed near the city of Ashkelon. There were no reports of injuries or deaths.