Amid softened tensions between Israel and the US after Prime Minister Benjamin Netanyahu's speech on Sunday, a familiar face is edging his way into the melee: Jimmy Carter. On Tuesday, the former president capped a week-long tour of the Middle East by meeting senior Hamas leader Ismail Haniyeh in the Gaza Strip.

Mr. Carter has been shunned in the past by both the Bush administration and Israeli leaders, who criticized his efforts to engage the militant Palestinian group that he says is crucial to any lasting Arab-Israeli peace. But analysts say Carter's ties with the more like-minded Obama administration, which has taken a firmer stand with Israel on some issues, may bolster his effectiveness as a regional peace broker.

"There is a big difference between Carter operating under Bush [and] Carter operating under Obama," says Alon Liel, a former Israeli Foreign Ministry director general. "His efforts had little value during the eight years of the Republicans. They have greater value now. He has access and connections with the leaders of [the] new America."

Ahmed Yousef, a senior adviser to Mr. Haniyeh, also acknowledges Carter's ties with Obama and potential to act as a go-between with the US, which considers Hamas to be a terrorist group. As a result, Obama's Middle East envoy, George Mitchell, has not met with Hamas leaders.

"He is close to President Obama and nobody in his type of position understands the conflict with all its problems like he does," says Mr. Yousef in a phone interview. "I think he will give Obama the information and analysis he needs to address this conflict in a proper way and to restore the image of America in the region after two decades of failed diplomacy."

Hamas reportedly thwarted two bombs targeting Carter's vehicle on Tuesday, according to Israeli newspaper Haaretz and news agencies.

Carter to Hamas: Accept US conditions for talks

On Tuesday in Gaza and last week in the Syrian capital, where Hamas chief Khaled Meshaal lives, Mr. Carter urged the militant group ruling Gaza to accept the conditions for talks laid out by the international community, including renouncing violence, accepting previous Israeli-Palestinian agreements, and recognizing the Jewish state's right to exist.

"I called on Hamas leaders that I met with in Damascus and I told Hamas leaders in Gaza today to accept these conditions," said Carter to reporters after meeting with Haniyeh for the first time. "They made several statements, and showed readiness to join the peace [process] and move towards establishing a just and independent Palestinian state."

Haniyeh, who welcomed the "new spirit" of the US as evidenced in Obama's June 4 Cairo speech, said Hamas will support a Palestinian state on 1967 borders, provided it would be under "full Palestinian sovereignty."

Why Israeli settler met with Carter

In Israel, whose government also considers Hamas a terrorist organization, Carter has on recent trips met with a cool reception – or no reception. But this time, he visited the Knesset and met with Israel's security cabinet. Even Shaul Goldstein, a prominent West Bank settler, agreed to meet him.

"Nobody in his position ever agreed to meet settlers. People won't meet settlers," says Mr. Goldstein, who heads the regional council of the Gush Etzion settlements in the Israeli-occupied West Bank.

"Carter is not the enemy," he says. "Maybe he's talking to the enemy. But Carter is not a terrorist, and he's not part of Hamas. The main goal is a dialogue, not a monologue. It is very important in the future to meet this kind of person."

In a surprising move, Carter said the settlements should be allowed to remain part of Israel.

'Not a giant step for mankind, Mideast peace'

The official charter of Hamas calls for the destruction of Israel, and it has provoked Israeli anger by firing rockets across the Gaza border and kidnapping an Israeli soldier, Cpl. Gilad Shalit.

Both Israelis and Palestinians see potential for Carter to facilitate a prisoner exchange. Mr. Shalit's parents reportedly gave him a letter to deliver to their son, and Hamas expressed hope that he could help free Palestinians.

"We are encouraging the talks to reach an honorable prisoners' exchange deal with Israel," said Haniyeh. "We welcome all the efforts ... in which Mr. Carter can help in order to reach a prisoners' exchange deal."

But foreign ministry spokesman Yigal Palmor downplays Carter's role. "It's not a giant step for mankind and for peace in the Middle East," he said, calling it a "humanitarian perk" Shalit and his parents.

The key, Mr. Palmor says, is to look at tangible results.

"It's very simple. Are there tangible results from bringing together people from different spheres and making them listen to each other?" he asks. "If the answer is yes, if you can point to concrete projects that have seen the light of day, and real improvement – however modest – in Israel-Arab relations, then this is a positive action. If there are no tangible results, then we should either be patient and wait, or conclude that these efforts are completely inefficient for the moment."

Hamas: 'Carter is the messenger we trust'

Carter, who said he was moved almost to tears by the situation in Gaza, promised to bring a "report" of the destruction he saw to Obama, as well as to Secretary of State Hillary Rodham Clinton and US Middle East envoy, George Mitchell.

"Someone as high-profile as Carter, coming to the region to meet with Hamas and the government of Ismail Haniyeh but also [Palestinian Authority] President [Mahmoud] Abbas, is very positive," says Mr. Yousef. "He can convey messages to President Obama about the situation in Gaza and in the West Bank and the consequences this blockade has had on our lives. Carter is the messenger that we trust – and that the world community trusts."

• Safwat al-Kahlout in Gaza City and Joshua Mitnick in Tel Aviv contributed reporting.