"Like I said about (U.S. President-elect Barack) Obama and (Republican presidential candidate John) McCain, here too we'll work with any elected (Israeli) prime minister," he told London's Arabic-language al-Sharq al-Awsat newspaper.
The three leading candidates to replace Ehud Olmert in February are Labor Party leader Ehud Barak, centrist Kadima Party Chairwoman Tzipi Livni and center-right Likud Party chief Benjamin Netanyahu.
Netanyahu, viewed by many as a hawkish conservative, has been assuring Arab, European and U.S. officials that if he is elected in February, as voter surveys suggest, he will continue talks with the Palestinians and govern with a broad coalition.
Abbas said he also "received a promise from Barak Obama twice -- once during the campaign and once when he visited me -- that the peace process would be a top priority."
Obama "promised he would not postpone the peace process in the Middle East," Abbas said. "When we spoke after the elections he said, 'I'm still committed to what I told you.'"
Abbas also said he planned to visit Washington before the end of President George W. Bush's term.
2014: The Year in Music [PHOTOS]