"It is unreasonable for the United States to not have a position and leave the parties to negotiate without an endpoint, we need a clear role from Washington, and we want to know if you are a mediator or a judge or facilitator of the negotiations."
Speaking to Al-Hayat newspaper, Yasser Abed Rabbo, an aide to the Palestinian president, said it made no sense to resume so-called proximity talks.
"The old way did not achieve any results and we will not accept a return to it," he said.
"We want to know, will the US side return to the old approach (of proximity talks or direct negotiations), or will they recognise a Palestinian state on the 1967 borders with Jerusalem as its capital?"
Clinton's speech on Friday came after weeks of fruitless US efforts to convince Israel to impose a second freeze on West Bank settlement activity.
A previous 10-month freeze expired shortly after the relaunch in early September of direct talks between Israel and the Palestinians, and Palestinian president Mahmud Abbas said he would not continue negotiations without a new building ban.
Washington offered Israel a package of incentives in exchange for a new three-month ban, but failed to win an additional moratorium.
Clinton said negotiations would continue in an indirect fashion despite the failure and called on both sides to "grapple with the core issues of this conflict: on borders and security, settlements, water and refugees, and on Jerusalem itself."
"We will work to narrow the gaps, asking tough questions and expecting substantive answers. And, in the context of our private conversations with the parties, we will offer our own ideas and bridging proposals when appropriate."
Palestinian political analyst Hani al-Masri said he was sceptical.
"It is clear from Clinton's speech that the US administration will aim to reach a framework agreement covering a temporary solution that will lead to a temporary state, which the Palestinian leadership has rejected," he told AFP.
US Middle East envoy George Mitchell was expected in the region on Monday, with Abbas due to meet him in Ramallah on Tuesday before heading to Cairo for a meeting with an Arab League follow-up committee on the peace talks.
On the Israeli side, reaction to Clinton's remarks was mixed, with one minister warning that the Jewish state would not discuss final status issues with the Palestinians "with a stopwatch in hand."
"It is neither logical nor in Israel's interest," Environment Minister Gilad Erdan told Israeli public radio.
Erdan said that Prime Minister Benjamin Netanyahu "will continue to work for peace with the understanding that its price will not be one that threatens Israel's existence and future."
But Labour party lawmaker and Social Affairs Minister Isaac Hertzog called on Netanyahu to present an Israeli peace proposal to Clinton immediately.
"The window of opportunity is very short both on the international and political levels," he told AFP.
On Sunday, Netanyahu made no comment on the state of peace talks during a weekly cabinet meeting, but his office distanced itself from remarks made by Defence Minister Ehud Barak at a conference with Clinton on Friday.
Barak said Jerusalem would one day be divided between Israel and a Palestinian state, prompting an official from Netanyahu's office to say the comments "were not coordinated with the prime minister."
"They represent the long-held views of the defence minister but don't represent the views of the government as a whole," he said.