Senior US officials said Israel's plans jeopardised the peace process and some even called the announcement's timing an "insult".
In an apparent move to defuse tensions, a statement from the office of Binyamin Netanyahu, Israel's prime minister, said "mutual confidence-building measures"were being considered, but no details of those measures were given.
Russia has joined other Quartet members in condemning Israel's settlement plans, but Neave Barker, Al Jazeera's correspondent in Moscow, said the Quartet's meeting was unlikely to win over sceptics.
"There's a tremendous amount of pessimism here in Russia about how effectively the Quartet can function as an organisation. Many here believe it runs contrary to what is actually going on in the Middle East," he said.
"[But] Russia has a privileged position as it were within the Quartet - it is the only member that continues to hold dialogue with all the various different warring factions within the region [including Hamas].
"To a certain extent there is level of leverage that Russia hopes it can use to bring those different factions the negotiating table."
The Quartet meeting on Friday will be attended by Ban Ki-moon, the UN secretary-general: Hillary Clinton, the US secretary of state: Sergei Lavrov, Russia's foreign minister; and Catherine Ashton, the EU's foreign policy chief.
Ban, who on Thursday repeated criticisms of Israel's plans to build more Jewish homes near East Jerusalem, said he hoped a meeting would advance peace efforts.
Underscoring the problems in getting peace efforts back on track, Ban said that indirect talks between Israel and the Palestinians alone were "not the best scenario".
"It is still worth a try," he told Russia's Ekho Moskvy radio, but he said that the proximity talks "should eventually lead to direct negotiations".
Lavrov said the Quartet was "determined to adopt a document that will be very specific and put forward a straightforward position that would reaffirm all the previous decisions of the international community".
The spat between Israel and the US has delayed a visit to the region by George Mitchell, the US special envoy to the Middle East.
PJ Crowley, the US state department spokesman, said later that Mitchell would meet both the Israelis and the Palestinians "at some point after the Quartet meeting" although it remained to be determined when.
"When we feel we have an understanding as to where both parties are and there's sufficient progress to justify meetings, we'll hold meetings," he said.