Peres for his part issued a statement saying he had called Abdullah to express "deep regret" for the shooting of Palestinian Jordanian Raed Zeiter at a border crossing, but the statement stopped short of quoting a full apology.
A Jordan palace statement said Abdullah "received a telephone call from Peres who, as president of the state of Israel, apologised for the martyrdom of Jordanian judge Raed Zeiter".
"Peres expressed his regrets for the incident and said he was deeply affected. He said Israel is committed to work together with Jordan on the probe into the incident."
Israeli troops on March 10 shot Zeiter, saying he attacked them, igniting a diplomatic row between Israel and the kingdom and sparking furore among Jordanians.
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The issue ignited a diplomatic row between Israel and the kingdom and caused furore among Jordanians.
Peres confirmed he had spoken to Abdullah today, but the wording of his statement left it unclear as to whether he had issued an apology.
"On behalf of the State of Israel, I wish to express my deepest condolences to the people of the Hashemite Kingdom of Jordan for the death of Judge Raed Zeiter at the King Hussein Bridge on March 10."
"As the President of the State of Israel I would like to express compassion to the bereaved family, (and) I share in their grief."
His statement did not mention whether Netanyahu had also spoken to the king.
Zeiter responded in kind and other soldiers joined the scuffle, with one of them firing three rounds at his chest, the PCHR said, citing "a witness".
Jordanian Prime Minister Abdullah Nsur has held Israel "completely responsible" for Zeiter's death and demanded an apology for the "hideous" killing.