However, Ayalon said it was not true he said sorry to Turki.
"Everyone who was present at the conference knows there was no apology from the deputy minister to the Saudi prince," said a statement from his office.
"To those who did not take part in the conference we suggest they check the video recordings to verify what really happened."
Earlier, Saudi Arabia's former intelligence chief and ex-ambassador to the United States said the handshake came after Ayalon publicly reprimanded him for not sitting together on a panel at the annual international security conference.
"I objected to sitting on the same panel with him not because he is the deputy minister of foreign affairs of Israel but because of his boorish conduct with the Turkish ambassador to Israel Ahmet Oguz Celikkol," Turki said.
In January Ayalon made a show of publicly humiliating Celikkol to demonstrate Israeli displeasure over a Turkish television show critical of the Jewish state.
Although Turki currently has no official government title he continues to carry out diplomatic work for the Saudi government.
Turki also said he objected to Ayalon's allegation that Saudi Arabia has not provided any aid to the Palestinian Authority. In fact Riyadh has given the PA hundreds of millions of dollars.
"Mr. Ayalon then asked me to come up to the podium to shake hands to show that there were no hard feelings," Turki said.
"I pointed to him that he should step down from the podium," Turki said.
"When we stood face-to-face, he said that he apologised for what he had said and I replied that I accept his apology not only to me but also to the Turkish ambassador."
Turki insisted that the handshake was in no way a step towards recognition.
"Until Israel heeds US President Barak Obama’s call for the removal of all settlements, the Israelis must be under no illusion that Saudi Arabia will offer what they most desire -- regional recognition."
----------------------------------------------------------------------