WASHINGTON (CNN) -- White House Deputy Chief of Staff Karl Rove should testify to Congress under oath regarding the U.S. attorney firings because he was "this close to be indicted" in the Valerie Plame CIA leak case, Senate Majority Leader Harry Reid, D-Nevada, said Thursday.

"When you're dealing with the cast of characters we have, you might consider putting people under oath," Reid said. "Karl Rove is an example -- who came within this close to being indicted for involvement in the Valerie Plame case. So I certainly think people should be under oath. At least Karl Rove"

Reid also said that with each passing day "it becomes more apparent" that the "White House is covering something up."

Asked if he meant a criminal cover-up, Reid responded, "there could be obstruction of justice charges in there someplace."

On the question of testifying under oath, Reid said beside Rove other potential witnesses might not have to swear to tell the truth.

"We don't have to have the same rule apply to everybody," he said

Rove's attorney Robert Luskin responded to Reid's comments in a written statement.

"Due process is not about pursuing the results you want," said Luskin. "It means respecting the results you get. Senator Reid should show a healthier respect for our system of justice than his unwarranted remarks reflect."