She will head back to Israel on Sunday to meet Prime Minister Benjamin Netanyahu and then brief the EU's 27 foreign ministers on the mission at a meeting in Luxembourg on Monday.
"I look forward to meeting Israeli and Palestinian leaders and encouraging them to seize the opportunity and engage in negotiations," Ashton said in a statement.
"With the momentous events going on in North Africa and following (US) President (Barack) Obama's speech last month, it is more urgent than ever that we kickstart the Middle East peace process."
"I have proposed a meeting of the Quartet to help relaunch negotiations and will be looking for positive signs from all sides," she added.
Her visit comes just days after she sent a letter to US Secretary of State Hillary Clinton, Russian Foreign Minister Sergei Lavrov and UN Secretary General Ban Ki-moon, a copy of which was seen by AFP.
The letter, sent on June 10, calls for "a gesture" before the summer from the diplomatic Quartet of Middle East peacemakers, which comprises the European Union, the United Nations, Russia and the United States.
In the document, Ashton calls for a meeting of Quartet principals "to adopt a statement" which would consolidate elements of Obama's call for negotiations to resume on the basis of 1967 borders with mutual land swaps, coupled with firm security guarantees.
"This is no time for unilateral moves on either side, since this could lead to escalation," she wrote in what appeared to be a reference to the Palestinians' plan to seek UN recognition for a state of their own this September.
Israel is fiercely opposed to the step, with Netanyahu looking to convince at least 30 countries to vote against the proposal, media reports said.
"I believe that what is needed now is a clear signal to the parties, and a reference framework that should enable them to return to the negotiating table," Ashton wrote.
"It is critical that we make a gesture before the summer, because we need to contribute to a calming of a volatile situation."
In Amman, Ashton meets counterpart Nasser Judeh before talks the next day with Israeli Foreign Minister Avigdor Lieberman and opposition Leader Tzipi Livni, followed by a West Bank visit to Palestinian Prime Minister Salam Fayyad and President Mahmoud Abbas.
Diplomatic sources in Brussels told AFP that Ashton hoped for a swift Quartet meeting, by early July at the latest, in Washington.
But Ashton "has limited room to manoeuvre," one source said, believing that Netanyahu "does not want to relaunch the negotiations and is busy campaigning for support against the recognition of a Palestinian state."
The question of recognising Palestine at the UN looks set to divide EU nations, with Germany for the moment opposed, Spain in favour and France not excluding recognition.
In Egypt, Ashton will attend a meeting of the Cairo Group, bringing together international organisations supporting a democratic transition in Libya.
The talks will be hosted by Arab League Secretary General Amr Mussa and bring Ashton together with the UN's Ban, by video-link, African Union commission president Jean Ping, and the Organisation of the Islamic Conference.
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