After Sunday's run-off election, Kwasniewski, 50, will be free to lobby for support to become the successor to Kofi Annan at the end of 2006, The Times of London said.
He has the full support of the United States after meeting with President George Bush Oct. 12, when Bush said Kwasniewski "has helped to advance the cause of peace by advancing the cause of freedom."
Under Kwasniewski, Poland broke with several of its EU partners, especially France and Germany, in sending troops to Iraq.
U.N. members operate a loose regional rotation in choosing a secretary-general. After an African head in Annan, most nations believe the next leader should come from Asia, which has held the post once before with U Thant of Burma from 1961 to 1971.
Russia has indicated it would use its veto power on the 15- nation Security Council to block an Eastern European, and China, another veto power, is also committed to appointing an Asian, the report said.