S. Secretary of State Condoleezza Rice to extend her regional visit.
Palestinians say relaxing Israel's border restrictions would help the region's ailing economy after the pullout of Israeli forces and Jewish settlers earlier this year, while Israel says measures are necessary to protect against terrorism.
But Ms Rice says she has new hope of a breakthrough after separate meetings with Israeli Prime Minister Ariel Sharon and Palestinian Authority President Mahmoud Abbas.
"If the parties work very hard... there is agreement in sight," Ms Rice said during a news conference with Mr Abbas, who also said a deal may be near.
Israeli officials had no immediate response.
The U.S. State Department announced that after the meeting Ms Rice delayed a trip to Asia and would return to Jerusalem Monday after meetings in Amman with Jordanian leaders.
She is then expected to continue working with Palestinians and Israelis on discussions centring on the Gaza border crossing at Rafa.
Israeli troops and settlers withdrew from Gaza in September after decades of occupation.
Sharon said the pullout and the withdrawal of Israelis from four small areas of the West Bank was part of a plan to reinvigorate the Israeli-Palestinian peace process.
The ease of movement for people and goods in and out of Gaza has been a central issue since Israel's historic pullout. Israel removed Jewish settlers in August, and the military pulled out in September.
Palestinians say they cannot build a viable economy without freedom of movement into Israel, across to the West Bank, and into Egypt on the southern side of Gaza.
But Israeli officials point out that terrorists have frequently used the border crossing to Egypt at Rafa to transport bombs and other weapons.
Israeli officials have said the "reality on the ground" -- meaning the extent to which militant groups refrained from attacks -- would determine the restrictions at the borders.
Discussions centre on Palestinians taking control of the Palestinian sides of the borders and the specific precautions they would take.
A plan under discussion would allow Palestinians to assume control for the Palestinian side of the border with Egypt with the possible involvement of members of the European Union.
Israel may be allowed to install closed-circuit cameras to help ensure militants do not use the border to import
It comes amid continuing violence, with Israeli security forces killing a leader of the Palestinian militant group Hamas in the West Bank town of Nablus, according to security officials form both sides.
Militants have launched rockets into Israel from Gaza in recent weeks, and Israel responded with aerial attacks on what the military called terrorist infrastructure, but overall the violence has been kept to a relative minimum.