Israel had approached the New Zealand ambassador in Turkey, who is also accredited to Israel, in the past few days, Clark told local radio on Monday.
"That diplomatic conversation has begun," she said.
But the New Zealand Herald newspaper quoted a senior government source as saying a draft "statement of regret" from Israel over the passport incident had been rejected by Wellington.
The draft had fallen short of the apology demanded by Clark, the source said.
Two suspected Israeli Mossad agents, Uriel Zoshe Kelman and Eli Cara, were deported in September last year after serving two months of six-month sentences for trying to illegally obtain New Zealand passports.
In June last year, when the two were sentenced, Clark said she had "no doubt" Kelman and Cara were agents.
She suspended high-level contacts with Israel, called off a visit by President Moshe Katsav and imposed visa requirements for Israeli officials.
The Israeli government has never publicly acknowledged the status of the two alleged spies.
Clark said Israel traditionally had friendly relations with New Zealand.
"It's our desire that that continue but clearly there is some discussion to go on," she said.
Australia expelled senior Israeli diplomat Amir Laty late last year with some reports linking him to Cara and Kelman.