"No system exists on overstays," Thompson said. "It's a problem."
In light of recent administration resignations, when asked about his intention to stay at DHS, Chertoff said, "Like everybody else in a Senate-confirmed position, I serve at the pleasure of the president, and so long as it pleases him to have me serve in this position, and of course, God willing, I'm happy to continue to do this job up until the very last day of the administration."
Attorney General Alberto Gonzales announced his resignation Aug. 27. His last day at the department is Sept. 17, and there has been wide speculation that Chertoff, a former assistant attorney general, U.S. attorney and federal judge, would be tapped to succeed Gonzales.
Asked about the possibility that he would be appointed to replace Gonzales, Chertoff said it would be "presumptuous" to speak for the president, but that he has "indicated what my intent is."
The Justice Department's Solicitor General Paul Clement has been named interim attorney general and will assume the post upon Gonzales' departure.
Clement will likely hold the position until the White House names a replacement and the Senate confirms that nominee.
Taking a shot at the turmoil of the Gonzales era at the department, Rep. Jane Harman, D-Calif., said to Chertoff, "Should you move to the Justice Department, I think you would spend a year and a half digging out of a very deep hole."
Though Chertoff is widely seen as a viable successor to Gonzales, the White House seems unlikely to move Chertoff, which would leave his post at Homeland Security open.