The Israeli prime minister has rejected Palestinian demands that Israel stop building its West Bank barrier as a condition for talks.
Mr Sharon has defended the building of the West Bank barrier
Ariel Sharon said "no condition shall be accepted" regarding the dismantling of the fence, reports said.
Palestinian Prime Minister Ahmed Qurei urged the US on Sunday to press
Israel to halt the barrier.
The US' Middle East envoy is reported to have begun a meeting with Mr Sharon following his talks with Mr Qurei.
The meeting is part of a continuing effort to give new momentum to the stalled Middle East peace process.
But the BBC's David Chazan, in Jerusalem, says the Israeli and Palestinian prime ministers are still poles apart and restarting the peace initiative will require more than just tactful diplomacy from the Americans.
Israel says it is building the barrier to stop suicide bombers but Palestinians say it is designed to annex land.
Mr Burns told Jordanian officials that the US is determined to move ahead with the road map peace plan.
The blueprint calls for the creation of a Palestinian state alongside Israel by 2005.
But Mr Burns said Palestinians must end violence against Israel.
He expressed hope that his meeting with Mr Sharon could pave the way for talks between the two prime ministers.
King Abdullah of Jordan - who hosted talks between Mr Qurei and Mr Burns - expects to visit US president George Bush on Thursday in Washington to boost peace moves.
Palace officials say Mr Qurei presented King Abdullah, a key US ally in the Middle East, with Palestinian views on how to resolve the peace stalemate with Israel.
Mr Qurei says there is no point in talking if Israel keeps building the wall
On Monday, Egyptian-mediated talks will begin in Cairo with Palestinian militant groups aimed at securing a ceasefire with Israel, Mr Qurei said.
Mr Qurei said he was ready to meet Mr Sharon, but the existence of the barrier would render talks meaningless.
"There is no point in holding any talks if the Israeli Government continues to build the wall," he told reporters on Saturday.
A UN report says Israel's partition is denying Palestinians access to land, hospitals and schools.
The US has signalled its disapproval of the barrier and of new Jewish settlements in the Palestinian territories by deducting almost $300m from loan guarantees - a rare financial sanction by Washington against Israel.