Antonio Guterres, who also is a former U.N. refugee agency chief, had more positive votes -- 11 -- than his 10 rivals for the position in Friday's nonbinding poll, diplomats told CNN on condition of anonymity. He also topped a similar poll in the 15-member council last month, diplomats said.
However, it may be significant that he also received negative votes -- two, diplomats said -- for the first time.
The straw polls are part of a process that is expected to end by this fall with the United Nations declaring a successor to Secretary-General Ban Ki-moon, whose second five-year term ends December 31.
Although there is technically no limit to the number of terms a secretary-general may serve, Ban is not seeking a third term, opening the door for the world body to have its ninth leader since its inception
71 years ago. No secretary-general has held office for more than two terms.
Perhaps by October, the Security Council will recommend a candidate to the 193-member General Assembly, which would approve the choice by consensus applause. But each of the Security Council's five permanent members -- the United States, Russia, China, the United Kingdom and France -- holds a veto, so those five must agree on a candidate before he or she is presented to the full assembly.
The council's straw polls, such as Friday's, are held in part to encourage a thinning of the field, with those not doing well perhaps getting the hint.
In Friday's poll, the council's 15 members were given choices for each of the 11 candidates: "encourage" (stay in the race), "discourage" (take a hint) or "no opinion."
Many potential major candidates drew several "discourage" votes Friday, diplomats said, which means countries are trying to signal candidates either to withdraw or prepare for a long slog of big power negotiating.
The next round of voting has not been scheduled.
Some other results from Friday's poll, according to diplomats:
• Vuk Jeremic, former Serbian foreign minister and the youngest candidate in the field, fared better than last month, receiving the second-most positive votes. He had "discourage" votes, though.
• Argentinian Foreign Minister Susana Malcorra also moved higher, to third place. She received six negative votes.
• Former Slovenian President Danilo Turk dropped from second place to fourth. He also received five "discourage" votes.
• U.N. development chief Helen Clark, from New Zealand, placed in the middle of the pack, but with eight "discourage" votes.
Eastern European nations feel it is their region's turn to produce a secretary-general, based on an informal global rotation pattern not defined in the U.N. Charter
Jeremic was the highest finisher from that zone.
It is still possible to have a new candidate who senses failure by the current 11 contenders or believes the large veto-carrying countries will look elsewhere.