The 74-year-old peace campaigner is being held with three other Western hostages by The Swords of Righteousness Brigade. Iraq's interior ministry said it had no information about the hostages.
The group threatened to kill all four on Saturday if its demands for all Iraqi prisoners to be set free from US and Iraqi jails were not met.
The Foreign Office in London said there had been no further developments.
Meanwhile, the body of Egyptian engineer Ibrahim Sayed Hilali was found, a day after he was seized by gunmen in Tikrit. His capture was separate to that of Mr Kember.
Mr Kember, from Pinner, north-west London, was seized in Baghdad on 26 November, with James Loney, 41, and Harmeet Singh Sooden, 32, both Canadians, and an American, Tom Fox, 54.
Mr Loney's younger brother, Ed, said yesterday: "We're hungry to hear something. I want them to realise that they really do have people that are on their side. The four [peace] workers have been working hard for the release of detainees in Iraq and that they want to continue that work."Reuse content