Mr. Netanyahu’s office refused to respond to Mr. Abbas’s comments. But the idea that Israel can rely only on its own military, not a third party, is a standard trope of the prime minister’s. He also says a fixed timetable is untenable, citing the volatility in the region. “Our attitude toward international forces is skeptical in the extreme,” said one senior Israeli official. “Timing can’t be artificial. It has to be based on performance, and we want to be able to judge what’s going on with performance.”

Jen Psaki, Mr. Kerry’s spokeswoman, said in an email that “there are many ideas being proposed from both the Israelis and the Palestinians, but it is premature to make any predictions about the final contents of a framework.” Others briefed on the negotiations said the secretary was trying to bridge the gap on security by pressing Mr. Abbas to extend his time frame, and by urging Israel to allow the United States, possibly with Jordanian involvement, to assess the conditions for withdrawal.

“The balance point has not been found yet,” said one Israeli security expert who has been consulted in the negotiations, speaking on the condition of anonymity because of the secrecy surrounding them. “The U.S. understands the Israeli position, accepts that there should be long-term presence, but looks for ways to reconcile between Israeli security needs and Palestinian needs for sovereignty and dignity.”

Mr. Abbas, 78, was relaxed and confident, if not quite optimistic, during the interview, sprinkling his politics with bits of humor. It took place in an outer sitting room where the Palestinian president has met delegations of left-leaning American Jews and foreign dignitaries and where, he recalled, the former American peace envoy George J. Mitchell said of the Israelis before departing in 2011, “They foiled me.”

He sipped sweet tea and then strong coffee, twice using a small buzzer to summon an aide who brought a single cigarette. He spoke in English, occasionally leaning on two colleagues for translation. (It took a few minutes to decipher whether Mr. Mitchell had said “fooled,” “failed” or “foiled” — Mr. Abbas joked that all three applied.)

On recognition of Israel as a Jewish state, Mr. Abbas said, “This is out of the question,” noting that Jordan and Egypt were not asked to do so when they signed peace treaties with Israel. He presented a 28-page packet he has been distributing widely that included a 1948 letter signed by President Harry Truman in which “Jewish state” was crossed out and replaced by “State of Israel”; statements by Israel’s founding prime minister, David Ben-Gurion; and a paper on Edwin Montagu, a Jewish member of the British cabinet who opposed the 1917 Balfour Declaration supporting a “national home for the Jewish people” in Palestine.