Muhammad's lawyers had appealed for the US Supreme Court to delay the execution while it examines a plea on whether his sentence was constitutional.
"The application for stay of execution of sentence of death presented to the chief justice and by him referred to the court is denied,'' Monday's Supreme Court ruling said.
Muhammad is now due to be put to death by lethal injection at 9:00 pm Tuesday in Virginia.
During the 2002 rampage that left 10 people dead and the Washington region paralyzed with fear, Muhammad, a skilled marksman, picked off victims in shopping malls, outside schools and at gas stations.
With the aid of a high-power sniper rifle and scope, he unleashed a reign of terror over three weeks, killing each of his victims with a single bullet fired from a distance.
But his lawyers argued Muhammad had not been properly legally represented during his trial in March 2004, when his legal team did not contest his request to defend himself even though he was clearly suffering from mental problems.
The defence team had asked the Supreme Court to rule on whether the 2004 sentence was constitutional, and a hearing had been set for November 24.
In a dissenting statement attached to the ruling, Justice John Stevens, backed by two other justices, said Monday: "This case once again highlights the perversity of executing inmates before their appeals process has been fully concluded.
"By denying Muhammad's stay of application we have allowed Virginia to truncate our deliberative process on the matter - involving a death row inmate - that demands the most careful attention.
"This result is particularly unfortunate in light of the limited time Muhammad was given to make his case to the district court.''
Muhammad's lawyers last week asked the nation's highest court to halt his execution to give the justices more time to examine an appeal against his death sentence.
"Trial counsel knew that Muhammad had been diagnosed with severe mental illness, and personally observed Muhammad's struggle with severe symptoms of psychiatric disorders prior to the start of the trial,'' said one of two documents presented to the court last week.
"These paranoid and delusional beliefs were clearly evident in and had devastating effect on, Muhammad's behavior in this case,'' the lawyers' petition read.
Muhammad's teenaged accomplice, Lee Boyd Malvo, who was 17 at the time of the shootings, is serving a sentence of life in prison.
Source: The Australian