Bin Hammam was slapped with the ban last month after he was found guilty of trying to buy votes in the FIFA presidential race by offering Caribbean football officials $40,000 each.
The former head of the Asian Football Confederation (AFC) has denied the charges, calling them "politically motivated", and is appealing the ban.
In a posting on his blog, Bin Hammam published a copy of a letter from FIFA secretary general Jerome Valcke, asking Caribbean Football Union members to come forward "with all relevant information" within two days.
Anyone who didn't come forward would "be subject to the full range of sanctions," the letter said.
"Based upon the evidence gathered to date, there is reason to believe that numerous people have not been completely candid and forthright in their statements about… whether they or others were offered or received money," it said.
Bin Hammam said he had been found guilty by a "kangaroo court - FIFA branch" and expressed outrage over the letter dated July 25, two days after the ban was imposed.
"Valcke, the co-owner of FIFA, issued a letter to the CFU member associations, threatening them to either admit that they have been bribed or he will open an investigation to find out if I have bribed them or not!!" he said.
The 62-year-old Qatari did not say how he obtained the document.
In an email to AFP, FIFA confirmed the letter was authentic but said Valcke acted within his role, supporting FIFA's investigation.
"I am surprised by these comments. Frankly, this is a bit low," Valcke said in the email.
"The facts clearly show that I acted at the request of the deputy chairman of the ethics committee and perfectly in accordance with the FIFA code of ethics. Contrary to what these comments appear to insinuate, I never deviate from my line of duty."
Bin Hammam's suspension by FIFA over the bribery accusations led to his withdrawal from the world football body's leadership election, handing incumbent president Sepp Blatter a fourth consecutive term in office.
The controversy has highlighted allegations of corruption in FIFA and sparked calls for reform of its governance structure, which anti-graft watchdog Transparency International earlier this month called "opaque".
The vote-buying accusations also raised questions over last year's controversial decision to award Qatar the right to host the 2022 World Cup, beating Australia, the United States, South Korea and Japan.
Bin Hammam, a former chairman of the Qatar Football Association, had played a key role in pre-vote lobbying for his home country's bid.