The official told the London-based Al-Quds Al-Arabi that Mitchell made the promise because Abbas refused to accept Prime Minister Benjamin Netanyahu's apology for declaring the approval of 1,600 new units during US Vice President Joe Biden's visit to the region.
Abbas demanded guarantees after Mitchell pressed him to accept the apology as a withdrawal of the construction plan.
According to the source, Mitchell vowed that Israel would halt settlement construction during indirect negotiations with the Palestinians.
He said the special envoy was likely to present the guarantee during his visit to the region next week, in which he is also expected to announce the launching of indirect negotiations between Jerusalem and Ramallah.
However the official stressed that the Palestinians would continue to reject the talks until the guarantee was explicitly presented to them. He added that US diplomats had recently held talks with Arab statesmen in order to press them to renew support for the talks.
MK Shaul Mofaz (Kadima) said Israel had made a mistake in announcing the construction in east Jerusalem during Biden's visit. "The US is Israel's best friend and strategic ally, so we must maintain relations with it," Mofaz said.
Speaking at a cultural event in Nez-Ziona he added, "We must not allow ourselves to embarrass the vice president and be publicly scolded by the secretary of state. It was Israel's political and strategic mistake."Meanwhile the US continues to criticize Israel for its construction plan in Ramat Shlomo neighborhood. A senior US official said Friday that Netanyahu's political standing is "perilous" because of divisions within his coalition over efforts to pursue peace with the Palestinians.
"The Israelis know the only way to stay on the positive side of the ledger – internationally and with us – is to not have them recurring," the senior Obama administration official said of the construction announcements.
Reuters contributed to this report