Israel and the Palestinians seemed closer Tuesday night than they have been for more than two years to a resumption of substantive peace negotiations, after both sides indicated satisfaction with an apparently American-brokered amendment to the Arab League peace initiative.

US Secretary of State John Kerry, who has spent recent weeks shuttling around the Middle East as well as meeting relevant players in Washington, also sounded fairly upbeat in comments Tuesday. He said there were still hurdles to clear, but “I don’t think you can underestimate… the significance of Saudi Arabia, Qatar, [United] Arab Emirates, the Egyptians, the Jordanians, and others coming to the table and saying, ‘We are prepared to make peace now in 2013.'”

The new optimism emerged on the very day that an Israeli father-of-five was stabbed to death by a recently released Palestinian security prisoner in the West Bank — the first such killing in more than a year, and an attack that was praised by Palestinian Authority President Mahmoud Abbas’s own Fatah faction.

A senior Israeli official told The Times of Israel on Tuesday night that the government of Prime Minister Benjamin Netanyahu was pleased with and welcomed the steps to advance the peace process taken by the Arab League, which on Monday said it could see a two-state solution based on minor adjustments to the pre-1967 lines, and by Kerry.

“Israel is ready to start negotiations — anytime, anywhere — without any preconditions,” the official said, “and expects the Palestinian side, also, to refrain from making preconditions. Both sides can present their positions at the negotiating table.”

Nonetheless, aides to Netanyahu privately cautioned that the path to a resumption of talks had not yet been completely smoothed. Although “very serious efforts” were under way, they said, it could be premature to anticipate an imminent resumption of negotiations, which broke off in late 2010.

In the prime minister’s circle, it was also stressed that new comments by Abbas, to the effect that he had “no preconditions” for a resumption of talks, should not necessarily be read as a breakthrough. Rather than dropping his longstanding demands for an Israeli settlement freeze and for Israeli agreement to negotiate on the basis of the 1967 lines, Abbas was likely merely making a semantic shift, branding such requirements as “obligations” rather than “preconditions.”

In Washington on Tuesday afternoon, Kerry hailed the Arab League’s shift. Speaking at a press conference with Spanish Foreign Minister Jose Manuel Garcia-Margallo, the secretary said the Arab initiative “never received the full focus and full attention and recognition” it deserved when it was first set forth by King Abdullah of Saudi Arabia.

He said Israelis had been asking him in recent days, “What are the Arabs going to do? What is the Arab attitude towards peace at this point in time? And so the Arab community — and I think they should be thanked for this — saw fit to come here to the United States as a delegation of the Arab League to make it clear that they are re-launching the Arab Peace Initiative.”

The secretary then set out what he said was the amended Arab League position: “Let me be very specific about what it does. Number one, if the Palestinians and Israelis reach a final status agreement between them, then the Arab community, 22 Arab countries and 57 Muslim countries that have signed up as members of the Organization of Islamic Cooperation, all of them have agreed, number one, that they would consider the conflict ended; number two, that they would establish the normalization of relations with Israel; number three, that they would enter into peace agreements with Israel; and number four, that they would provide security for all states in the region. In other words, they are offering a security arrangement for that region.”

Added Kerry: “This is literally a statement by the Arab world that they are prepared to make peace providing the Palestinians and Israelis reach a final status agreement. I don’t think you can underestimate… the significance of Saudi Arabia, Qatar, [United] Arab Emirates, the Egyptians, the Jordanians, and others coming to the table and saying, ‘We are prepared to make peace now in 2013,’ but one more thing: Unlike the agreement, the proposal that was put forward originally which only talked about ’67 lines, nothing else, yesterday they stated that they are prepared to accept ’67 borders with adjustments to reflect mutually agreed-upon land swaps, recognizing some of the changes that have taken place. That is a very big step forward.”

Kerry quoted Israeli Justice Minister Tzipi Livni, who hailed the shift early on Tuesday. Livni, he noted, said that the new Arab League position “sends a message to the Israeli public that this is not just about us and the Palestinians.”

The secretary said “We have a lot of homework to do, a lot of tough hurdles to get over, but each step forward is the way you get there. And the old saying, ‘The journey of a thousand miles begins with a single step.’ Well, President Obama began that step when he went to Israel [in March], offered a vision for peace. He instructed me to continue that work. We’re taking more steps. Yesterday was another step. And we’re going to continue to march forward and try to bring people to the table despite the difficulties and the disappointments of the past.”

Speaking on behalf of an Arab League delegation to Washington, Qatari Prime Minister Sheik Hamad Bin Jassem Al Thani on Monday had called for an agreement between Israel and a future Palestine based on the pre-67 lines. But, unlike in previous such proposals, he cited the possibility of “comparable,” mutually agreed and “minor” land swaps between the Israelis and the Palestinians.

Al Thani spoke after his delegation met across the street from the White House with Vice President Joe Biden and Kerry, who has been pushing Arab leaders to embrace a modified version of their decade-old initiative as part of a new US-led effort to corral Israel and the Palestinians back into direct peace talks.

Palestinian chief negotiator Saeb Erekat gave backing to the softened Arab League stance, saying that a plan including minor land swaps was consistent with the Palestinian Authority’s official position.

“In the event that Israel should accept a two-state solution based on 1967 borders, the Palestinians may consider small border adjustments, as long as it does not harm Palestinian interests,” Erekat said.

That stance was criticized by Hamas and by some in Fatah, including former PA minister Nabil Sha’ath. Hezbollah and Islamic Jihad also criticized the Arab League’s shift.

The Arab League’s statement was welcomed by a number of Israeli officials, who called to seize the opportunity and revive negotiations as soon as possible.

“This is an important step for the Arab world that has a chance of being a groundbreaking move and should be viewed seriously,” Labor Chairwoman Shelly Yachimovich said Tuesday night. Yachimovich urged Netanyahu to respond positively to the Arab League’s proposal and declared that the Labor Party would offer its support for significant steps towards an agreement.