For Qatar, the outreach to Gaza also is part of far wider ambitions to become a major policy-shaper in the Middle East. The tiny Gulf nation has emerged as a strong backer of the Muslim Brotherhood, which has risen to power in Tunisia and Egypt after the fall of those countries' autocratic leaders in early 2011.
In February, Qatar brokered talks between Hamas' Mashaal and his longtime rival, internationally backed Palestinian President Mahmoud Abbas.
Qatar has also sought to influence Syria's rebels. This month, it hosted Syrian opposition groups in a breakthrough effort to unite rebel factions under one coalition, which has opened the way for greater international recognition and promises of aid. Qatar had led calls to supply Syrian rebels with heavy weapons to counter air and tank attacks by Bashar Assad's forces. Qatar also was a key backer of the Libyan uprising.
On Thursday, Qatar also invited the newly formed Syrian opposition coalition to appoint its ambassador to the Gulf state, the Qatari news agency reported.
Salman Shaikh, director of The Brookings Doha Center in Qatar, said it's not a matter of "having to go all for Iran or all for Qatar.'
"What Qatar is trying to do is change the reality. They are trying to blaze a trail that will weaken the international isolation of Gaza from the Israeli blockade," he said.
Qatar has so far stopped short of offering any kind of military support for Gaza to avoid a rift with Washington, Israel's most powerful ally.
Iran takes a very different view.
Officials in Tehran boasted this week about its longtime arms support for Hamas militants — part of Iran's bookend strategy to equip anti-Israel factions in Gaza and Hezbollah in Lebanon on Israel's northern border.
Gen. Mohammad Ali Jafari, the head of the Revolutionary Guard, said Iran supplied fighters in Gaza with the technology to "quickly" produce the Fajr-5 missiles. The statement fits with previous Iranian denials that it is not directly sending missiles to Gaza, but suggests close coordination on construction and movement of supplies, presumably through the smugglers' tunnels linking Gaza and Egypt.
The Gaza fighting, at the least, bought Iran some restored street credibility as its image was battered by its backing for the regime of Syrian President Bashar Assad.
"The war with Israel reminded Hamas that Israel is the main issue not Syria," said Heshmatollah Falahatpisheh, a professor in politics in Tehran's Allameh University.
But Hamid Reza Shokouhi, editor of Iran's independent Mardomsalari daily, said the bump in Iran's popularity with Gazans could be only "short-term." He questioned whether Iran could sustain its influence in Gaza against the almost unlimited resources of Qatar and its Gulf partners.
Iran's Foreign Ministry and a parliamentary committee have applied for permission to visit Gaza in the coming weeks via the border crossing with Egypt, said Hasan Qashqavi, deputy foreign minister in charge of consular affairs.
Qatar's prime minister said the competition for influence is Gaza only likely to intensify as other nations such as Turkey and Egypt reach out.
"We are not trying to take Hamas from anybody else, from Iran or others," he said in the CNN interview. "Hamas, they have to decide for themselves. I think they are pretty mature to decide for themselves."
Associated Press writer Nasser Karimi in Tehran, Iran, contributed to this report.
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