The Palestinians say they will not enter indirect peace talks unless Israel scraps the new settlement plans.
Clinton met her Russian counterpart Lavrov, the U.N.'s Ban, EU foreign policy chief Catherine Ashton and Quartet Representative Tony Blair over a closed dinner on Thursday evening before Friday's formal meeting. No details of that meeting were disclosed.
The quartet was formed in 2002 in Spain to assist in mediating an end to escalating violence in the Israeli-Palestinian conflict. It last met on the margins of the U.N. General Assembly in September.
But its achievements so far have been meagre, leading some analysts to dismiss it as an expensive club for diplomats.
Moscow had originally hoped to organise a full-scale international conference on the Middle East this year but the lack of progress on Israeli-Palestinian peace talks forced Russia to settle instead for hosting a quartet meeting.
(Writing by Michael Stott; Editing by Jon Hemming)