"Right now Qatar is the main problem and definitely not part of the solution," he wrote in an email. "The ruling family in Qatar should understand that this is a dangerous game their emir is playing."
But in a development reflecting both Qatar's significance and influence over Hamas, the Gulf country's news agency reported that Abbas arrived Wednesday in Doha, where he was due to hold talks with Mashaal and the emir.
It is hardly the first time Qatar has been accused of taking an unpopular stance in the region.
Saudi Arabia, the United Arab Emirates and Bahrain recalled their ambassadors from Qatar in March, saying it failed to uphold its end of a security agreement to stop meddling in other nations' politics and backing groups threatening regional stability. Analysts widely saw that as a rebuke of Qatar's support for Islamist groups and its activist foreign policy, including its backing of the Al-Jazeera satellite network, which has nettled governments across the region.
Qatar's leaders reject suggestions that they are behind Hamas, and insist that the Gaza funding is intended for those who live there.
"Qatar does not support Hamas. Qatar supports the Palestinians," Qatari Foreign Minister Khalid al-Attiyah told CNN in late July.
The former Qatari emir, Hamad bin Khalifa Al Thani, has at least publicly attempted to promote reconciliation between Hamas and the Western-backed Palestinian Authority that governs the West Bank. He brokered an interim unity government between Abbas and Hamas in early 2012, but that was never implemented.
Before the year was out, the emir traveled to Gaza, becoming the first head of state to visit the seaside territory since Hamas militants seized control in 2007. He launched more than $400 million worth of projects, including plans for housing, a hospital and roads, and called for Palestinian unity.
Khalil Shaheen, a political analyst in Ramallah, suggested the idea that Qatar is solely in Hamas' camp is overblown. He said it has also provided funding for Abbas' government and has not tried to tie its Gaza aid to Hamas' military activities.
"There never was a real crisis between Qatar and the Palestinian Authority even during the worst times between Fatah and Hamas," Shaheen said.
He said Qatar wanted a role in the ceasefire talks based on its good relations with Hamas and to show that Egypt is "not the only dominant player in the region."
For the U.S., Qatar plays a role that it often can't by acting as a go-between with groups deemed unsavory by Washington. It earlier this year brokered the release of U.S. Army Sgt. Bowe Bergdahl in exchange for five Taliban operatives in Afghanistan.
U.S. State Dept. spokeswoman Marie Harf described the Qataris as "a key partner" in the effort to forge a peace deal in Gaza earlier this week, before talks collapsed. Responding to questions about whether they support terrorism and Hamas, she said they play a key role in getting Hamas to agree to a cease-fire.
"We need countries that have leverage over the leaders of Hamas who can help get a cease-fire in place, and Qatar certainly plays that role," she said.
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Associated Press writers Mohammed Daraghmeh in Cairo, Josef Federman in Jerusalem, Karin Laub in Gaza City, Gaza Strip, and Abdullah Rebhy in Doha, Qatar, contributed reporting.
