Sharm El-Sheikh, Egypt (CNN) -- The arduous Middle East peace talks continued on Tuesday in Egypt, where officials from Israel and the Palestinian Authority "have begun a serious discussion on core issues," a top U.S. diplomat said.
"They have agreed to begin first on working to achieve a framework agreement for permanent status. That work is now well under way," said U.S. special envoy George Mitchell.
Israeli Prime Minister Benjamin Netanyahu and Palestinian leader Mahmoud Abbas attended the sessions, along with Mitchell and U.S. Secretary of State Hillary Clinton. Officials met for an hour and 40 minutes in bilateral and multilateral meetings.
Mitchell briefed the media after the session and said the goals remain a two-state solution, condemnation of violence and working toward security, and a resolution of all issues. He reiterated the Obama administration's position that Israel should extend the moratorium on settlement construction in the West Bank, a hiatus scheduled to end later this month.
"We believe we are moving in the right direction overall," he said.
The parties are aiming toward resolving all core issues within the next 12 months in a process that kicked off recently during a meeting between Netanyahu, Abbas and Clinton in Washington. Issues separating both sides include Jewish settlements in the West Bank, the future of Palestinian refugees, Israeli security, and the status of Jerusalem.
Clinton and Mitchell are expected to continue talks with the leaders in Jerusalem on Wednesday.
Tensions are growing over the issue of possible new Israeli settlements in the West Bank. Netanyahu is under pressure from the Palestinians and the Obama administration to extend a 10-month moratorium on building Israeli settlements in the disputed West Bank territory. That moratorium is set to expire September 26.
Palestinians have said the construction would torpedo the talks, but Israel says some construction is likely.
Acknowledging that the settlement issue is sensitive in Israel, Mitchell said both sides have a responsibility to continue the meetings and he called on Abbas to take steps to advance the talks.
"All issues ultimately must be resolved by the parties themselves," Mitchell told reporters. "The United States will, as we've said on many occasions, be an active and sustained partner throughout the talks and will, when necessary and appropriate, make proposals and provide encouragement to the parties. In the end, matters must be resolved by parties themselves and we hope and expect that they will do so."
Mitchell said he wouldn't divulge many details about the talks, noting the importance of confidentiality and sensitivity. But he did say "our vision is for a two-state solution."
"That includes a Jewish, democratic state of Israel living side by side in peace and security with a viable, independent, sovereign and contiguous state of Palestine. But of course, this is one of many sensitive issues the parties need to resolve themselves and that's the point of negotiations. The parties will reach agreement on all major issues."
Prior to arriving in Sharm el-Sheikh, Egypt, on Tuesday morning, Clinton said if there are no negotiations, there will be no security for Israel and no state for the Palestinians.
This isn't Clinton's first time to participate in an attempt to secure a two-state solution.
As first lady, though not a principal negotiator, she traveled to the Middle East to meet with Israeli and Palestinian leaders in support of President Bill Clinton's policies on the issue. The president eventually hosted Israeli and Palestinian leaders at Camp David, Maryland, for what proved to be unsuccessful final-status talks.
Now, as secretary of state, Clinton has a second chance and a more direct platform to help the parties reach a comprehensive settlement.
Should Clinton help shepherd an agreement, it would "fulfill a longtime desire to succeed in this area," said Ned Walker, who was U.S. ambassador to Israel during part of President Clinton's second term.
"She has something of a long history of being involved" in the region, said Walker, who also used to be an assistant secretary of state for Near Eastern affairs. "This would be a great fulfillment for her personally and for the administration and the country."
CNN's Ben Wedeman, Jason Hanna and Laurie Ure contributed to this report.