Rice might even join the talks late on Monday night between Israeli Defence Minister Shaul Mofaz and Palestinian Civil Affairs Minister Mohammed Dahlan, the official added, asking not to be identified.
Earlier on Monday, Rice decided to delay her departure to Asia while talks continued to nail down an agreement on reopening the Rafah terminal, according to her spokesman.
Later she flew to Jordan to pay her respects to the victims of Wednesday's triple bomb attacks in Amman.
Aljazeera's correspondent in Palestine reported that the talks conducted by Rice with Palestinian and Israeli officials had failed to produce a breakthrough.
State Department spokesman Sean McCormack had said Rice would fly to Jordan to pay a condolence call for the hotel bombings, but return to Jerusalem rather than continue on to Pusan for an Asian regional conference.
"We're still working the issues with both sides," McCormack said of efforts to reopen the Rafah terminal, a key transit point between Egypt and Gaza which has remained largely closed since Israel's pullout from the Gaza Strip, completed on 12 September.
Before news of the failure was confirmed, Palestinian President Mahmoud Abbas had struck an optimistic note after talks with Rice, saying an Israeli-Palestinian deal on reopening the Gaza-Egypt border was imminent.
Verge of agreement
"We have discussed issues relevant to the Gaza Strip in order to avoid transforming the Gaza Strip into a big prison and we spoke of the necessity of opening the Rafah crossing point," Abbas said.
"We are on the verge of reaching an agreement on this issue," he said, flanked by Rice after talks in Ram Allah.
Rice added: "It is very important for ordinary Palestinians ... that there be freedom of movement established between Gaza and the West Bank."
Rice (L) met Sharon (R) during
her trip to Israel on Sunday
Speaking about the Rafah issue, Rice told the news conference: "I believe with some will and creativity, an agreement ... should be within sight."
Rice later decided to extend her visit to the Middle East region to try reaching agreement on Rafah.
Aljazeera's bureau chief in Ram Allah, Walid al-Umari, said the Rafah issue was the sole focus of Rice's meetings with officials in both Israel and the West Bank.
There are also continuing talks between the Palestinian and Israeli sides, namely between Mofaz and Dahlan, which are being attended by the international peace Quartet envoy James Wolfensohn.
Rice had hoped to cross over to the Palestinian territories with an agreement in hand after meeting the Israeli side, but the meetings had failed to achieve a breakthrough, al-Umari said.
According to al-Umari, differences on the Rafah issue are mainly over:
Israel's demand for retaining the right to veto individuals entering or leaving the Gaza Strip, those entering it in particular. Palestinians refuse to agree to such a condition.
Operation of the joint control room in which Israel is to take part besides European observers. At the crossing itself, only Palestinian and Egyptian officials are supposed to be in charge.
Israel's demand for veto power and direct monitoring of the crossing using video cameras. Palestinians oppose this on the grounds that it might cause 24-hour delays at the crossing.
Moments before the news conference began, Dahlan said an agreement had been reached.
"We have reached an agreement on the subject of reopening and operating the Rafah terminal," he told AFP.
"The need to deal with the infrastructure of terrorism is a road map obligation"
US Secretary of State
Chief Palestinian negotiator Saeb Erikat said the Palestinians had accepted a compromise solution put forward by Wolfensohn.
"The Americans are now waiting for Israel's agreement," he said.
On Sunday, on her way to Israel from Saudi Arabia, Rice mentioned the changed environment in Israel and the Palestinian territories, with the death of Palestinian president Yasser Arafat a year ago and Israel's withdrawal from the occupied Gaza Strip.
"A Palestinian state would indeed enhance Israeli security," Rice said.
Asked what she would request of the two sides during her trip, Rice said that the Palestinians needed to tackle fighters ahead of January's parliamentary election and that a democratic state could not be built when resistance groups remained armed.
"The need to deal with the infrastructure of terrorism is a road map obligation," she said referring to a US-backed peace plan setting out steps for a Palestinian state alongside a secure Israel.
Rice said she would make clear to the Israelis that they had to be aware of the consequences of their reaction. "The Israelis have very important road map obligations and we will talk about that, too," she said.
Habit of cooperation
"Israel should do nothing to pre-judge final status or the outlines of a final settlement. The parties have now had the habit of cooperation in the Gaza withdrawal, and it is in our hope that they are going to continue to build on that," Rice said.
Israel has failed to meet its road map commitment to freeze Jewish settlement building in the West Bank.
Rice met Saudi King Abdullah and
discussed the issue of terrorism
Rice also said the Bush administration was under no illusions about the difficulty of spreading democracy in the region.
"We are not naive about the pace, or difficulty, of democratic change," Rice said, "but we know that the longing for democratic change is deep and urgently felt."
Profound change is under way in the Middle East, Rice said near the close of a diplomatic trip that began with encouragement for incipient democracy in post-Saddam Iraq and will end on Monday with condolences for nearly 60 people killed in a bombing last week in Jordan.
"We have hope for peace today because people no longer accept that despotism is the eternal political condition of the Middle East," Rice said.
Rice gave cautious encouragement to lesser changes and reforms in Saudi Arabia and Egypt in Sunday's address.
Rice met Saudi leaders earlier on Sunday in Jedda and said that although Saudi Arabia could do more to root out the sources of terror financing, the two countries shared commitment to fight terrorism.
On her flight from Saudi Arabia to Israel she renewed criticism of Syria for dragging its feet in cooperating with a UN investigation into the killing of former Lebanese prime minister Rafiq al-Hariri in Beirut on 14 February.
Syria President Bashar al-Assad
made a defiant speech last week
"We have to say the Syrians have not yet cooperated," Rice said, dismissing Syrian complaints about the inquiry and its plans to perform its own investigation.
"That's just not going to cut it," she said.
In Israel, Rice met Prime Minister Ariel Sharon and then Palestinian President Abbas in the West Bank.