The White House had asked the appeals court to issue a stay preventing Lindsey from testifying until the Supreme Court decides whether to intervene in the matter.
Supreme Court officials say the Clinton administration's emergency request for an order to block the testimony immediately went to Chief Justice William Rehnquist. He could decide the issue himself or send the matter to the full court.
Lindsey is the president's closest confidant and a White House lawyer. Lindsey argues that because he's a lawyer, conversations he has with the president should be confidential.
Meanwhile at the White House, officials are holding their breath waiting for word on the outcome of tests on Monica Lewinsky's dress, CBS News Senior White House Correspondent Scott Pelley reports.
The FBI is conducting the tests to determine whether, as Lewinsky claims, a stain on the dress contains Mr. Clinton's semen.
Sources tell CBS News that Lewinsky will not be appearing before the grand jury on Tuesday, but when she does, her testimony may take as little as one day.
In the meantime, Mr. Clinton is preparing for his date before the grand jury.
A White House spokesman said Monday that the president "said he told the truth and that he will continue to tell the truth." But the fact is, virtually no one here knows what Mr. Clinton is planning.
In a sense, the president has become estranged from the White House. He has been isolated by his attorneys, David Kendall, Mickey Kantor and White House counsel Charles Ruff.
It was they who virtually silenced the president on this issue, who hid Mr. Clinton's subpoena from top aides and told the public Mr. Clinton's testimony would be taped when in fact the grand jury will see him live.
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Senior aides find themselves publicly defending and privately doubting. On Monday, one senior advisor began a conversation by saying, "I believe him." Then, this same advisor lamented, "The president does not feel free to turn to any of us and say, 'Here are the facts, here's what happened.' He's isolated ... we can't give him advice on this."
On Monday, Ken Starr spoke to reporters for the first time since the president agreed to testify.
"We have tried and we continue to try to gather all the fact, not some of the facts, but all of the facts," Starr said.