Setting preconditions for talks, the PA will give the USA negotiating team, arriving this week in the Middle East in an attempt to revive the peace process, 45 days to support their terms for a two-state solution.
Also present at the Tuesday meeting were Jason Greenblatt, Mr Trump's envoy for negotiating Israel-Palestinian peace, US deputy national security adviser Dina Powell, and Saudi Arabia's new ambassador to the US, Prince Khaled bin Salman.
On Wednesday, the United States delegation is expected to arrive in Israel, the media outlet said citing a White House aide.
Trump has asked his delegation to focus the talks on this trip around several broad themes, including finding "a path to substantive Israeli-Palestinian peace talks, combating extremism [and dealing with] the situation in Gaza, including how to ease the humanitarian crisis there", according to a senior White House official. However, no specific date was announced.
Netanyahu will be in Russian Federation this week to meet with Kremlin President Vladimir Putin.
"I have met with Trump envoys about 20 times since the beginning of his term as president of the United States", Haaretz quoted Abbas as saying, citing "notes taken by some of the meeting's participants". "They said they would consider it but then they didn't get back to me", Abbas said.
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Abbas is demanding that Trump publicly declare the two-state solution as the only solution and that he insist Israel "stop creating facts on the ground" through construction in Jerusalem, Judea and Samaria.
Abbas recently griped that he doesn't understand the Trump administrations policy toward Israel - complaining that the administration won't go public with its support for a two-state solution and a halt to Israeli settlements.
"We hope they bring clear answers this time", he added.
"I don't even know how they are dealing with us, because his entire administration is in chaos", Abbas told a delegation of dovish Israeli lawmakers visiting Ramallah.
Mr Kushner, the USA president's son-in-law, was charged with helping to broker a deal between Israelis and Palestinians after Mr Trump took office.
Netanyahu's hand has been weakened by an ongoing corruption probe.