(Adds Gillerman's explanation of the encounter, paragraphs 5-6)

By Claudia Parsons

UNITED NATIONS, July 24 (Reuters) - Whether it was a chance encounter, an ambush at the men's room or a staged first meeting, the U.N. ambassadors of Israel and Syria walked side by side and exchanged words for the first time on Tuesday.

Israeli Ambassador Dan Gillerman told reporters he had shaken hands and spoken with his Syrian counterpart Bashar Ja'afari when the two met in a hallway outside the U.N. Security Council on Tuesday.

"That's the first time we spoke," said Gillerman, who has been Israel's ambassador since the start of 2003. The two were seen walking from near the men's room outside the Security Council into the antechamber of the council.

Asked whether they had discussed anything more significant than the weather, Gillerman said, "We did not discuss the weather." Asked if they had shaken hands, he said, "Yes."

Gillerman later told Reuters that he was taking a group of Israeli school children on a tour of the United Nations when an Israeli journalist accompanying them stopped to interview the Syrian ambassador.

But the reporter had identified himself as a Romanian rather than an Israeli, and when Gillerman learned of this deception, he quickly sought out Ja'afari and apologized for the journalist.

Ja'afari, however, denied he had spoken to Gillerman.

"I wasn't walking with him ... he was following me," he told reporters.

"I didn't address him, I didn't talk with him." Asked if he had shaken hands, Ja'afari said, "Neither hands nor feet."

The encounter took place at a spot where reporters gather to speak to diplomats going in and out of the council.

Talks between Syria and Israel collapsed in 2000 without resolving the fate of the Golan Heights, a plateau occupied by Israel in the 1967 Arab-Israeli war, and annexed in 1981 in a move not recognized internationally. Both sides have signaled a readiness to resume talks in recent months.

In remarks broadcast on Saturday, Israeli Prime Minister Ehud Olmert said Syria was setting an "impossible threshold" for peace talks by demanding Israel commit to withdraw fully from the occupied Golan Heights before negotiations resume.

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