If confirmed, critics could seize upon the move as a sign that he has given up hope of finding banned arms.
"He has told the DCI (Director of Central Intelligence George Tenet) that he doesn't want to go back, they have been trying to get him to stay," the source told Reuters on condition of anonymity.
It was unclear whether the CIA had had any success in persuading Kay, who came back to the United States for the Christmas holidays, to stay on the job, the source said.
A CIA spokesman declined to comment. Kay, reached earlier this week, also declined to comment and referred questions about his status to the CIA.
Tenet last June appointed Kay, a former UN weapons inspector, as a special adviser to lead the search for biological and chemical weapons and any signs of a resurrected nuclear weapons programme in Iraq.
But the hunt, which is being conducted by the Defense Department's Iraq Survey Group, has come up empty, finding no stockpiles of biological and chemical weapons or any evidence that Iraq had restarted a programme to develop nuclear weapons.
The Bush administration cited weapons of mass destruction as its main justification for the war against Iraq that ousted Saddam Hussein from power last April.
Alleged Iraqi WMDs were Bush's
main justification for war
A US official, who also spoke on condition of anonymity, called Kay's status "up in the air".
US officials last month said Kay had told administration officials he was considering leaving the job as early as January, citing family obligations.
At that time, officials described Kay as frustrated that no banned weapons were found and that some of his staff had been diverted to other tasks.
The White House also said the weapons hunt was a priority for the administration whether or not Kay stayed on the job.
The Carnegie Endowment for International Peace last week issued a report that accused the Bush administration in the lead-up to the war of making the threat from Iraq sound more dire than the underlying information warranted.
The report's authors said they did not expect any large stockpiles of biological and chemical weapons to be found.