Sources close to Fayyad confirmed the report.
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Abbas was due to return to the West Bank from Jordan on Thursday, and it was not immediately clear whether he would accept the resignation of the US-educated economist.
According to rumors, Abbas wanted to sack Fayyad following internal political wrangling.
Western governments have offered staunch support to Fayyad ever since he became prime minister in 2007, seeing him as the architect of Palestinian state-building efforts, and his departure could complicate their ties with Abbas.
Long-strained relations between the 61-year-old Fayyad and Abbas worsened last month when the prime minister accepted the resignation of his finance minister, against the wishes of the president.
Initially successful in revitalizing a sluggish Palestinian economy, Fayyad ran into trouble last year when Israel and the United States withheld vital funds to punish the Palestinians for seeking de facto statehood at the United Nations.
They said the unilateral move ran counter to previous accords and their financial penalties meant Palestinian public sector salaries went unpaid, stoking street protests.
Abbas's Fatah party accused Fayyad of failing to foresee the turmoil and the party's council issued an unprecedented rebuke last week, saying: "The policies of the current government are improvised and confused in many financial and economic issues."
When he visited the West Bank last month, US President Barack Obama explicitly described both Abbas and Fayyad as "true partners" in a peace process that he hopes to revive between Israeli and Palestinian leaders in the coming months.
US Secretary of State John Kerry also met Fayyad this week during a visit to Jerusalem in an apparent show of support for the beleaguered prime minister.
Kerry spent much of his visit to Ramallah trying to work out a way to keep Fayyad as premier. During his visit, he stressed the need to strengthen the Palestinian economy signaling that Washington wants Fayyad to continue to head the government.
Speaking earlier on Wednesday about the rumors of a division between Fayyad and Abbas, a senior diplomat in Jerusalem said Western aid donors would be very upset to see the respected prime minister leave his post.
"Fayyad's departure would have a serious impact on relations with the international community," said the diplomat, who spoke on condition of anonymity. "It is hard to overstate how important Fayyad has been."
The diplomat added that Fayyad's institution-building drive in the West Bank had been "the single best thing" that had happened in the Palestinian territories in recent years, adding that the premier was also highly trusted by Israeli leaders.
Fayyad's close ties with the West have irritated senior Fatah officials, who have accused him of trying to build an unassailable powerbase, despite the fact that he had no significant political support amongst ordinary Palestinians.
The Islamist group Hamas , which controls the Gaza Strip, has repeatedly accused Fayyad of helping Israel to blockade its coastal territory and has never recognized him.
Fayyad spent two days in hospital earlier this month after suffering severe stomach pains. He had a heart attack in 2011.
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