Bigley, a 62-year-old engineer working at a US military base, was seized 18 days ago with two Americans.
The Americans were killed by their captors, al-Tawhid wa al-Jihad group led by Abu Musab al-Zarqawi, an ally of al-Qaida.
"I am getting communiques from dear friends of mine - business friends and personal friends - who are based in Kuwait ... that Ken possibly, and I reiterate possibly, has been handed over from the political baddies to the regular baddies," Paul Bigley told Sky News on Monday.
He said he had also heard that those now holding his brother would "like to negotiate a financial settlement".
He said his information had originally come from a newspaper in Kuwait, which on Saturday reported that an Iraqi armed group was prepared to negotiate Bigley's release.
The same newspaper accurately predicted the release of two Italian aid workers last week.
No further information
A spokesman for Britain's foreign office could give no further information on the situation. "We have seen those reports and we are looking into them," he said.
To the agony of his family, video footage of a distressed-looking Bigley was released last week showing him chained and squatting in a cage. He chided UK Prime Minister Tony Blair for not negotiating with his captors.
"Ken possibly, and I reiterate possibly, has been handed over from the political baddies to the regular baddies"
Blair has insisted his government will make no deals, but has appealed for the captors to make contact.
Paul Bigley, who lives in Amsterdam, said he was now feeling more positive about the prospects for his brother.
"I am ... hanging on to this thread, hoping that it is true. If this horrible thing can be put to bed with a financial settlement then at least we are away from the political side of things which is absolutely horrible.
"We are not stopping until we get Ken home," he added.
Al-Tawhid wa al-Jihad can be understood to mean the unity of God and the holy struggle.