Prosecutors are expected to present a parade of family members during the proceedings.
Moussaoui claims he was a 9/11 co-conspirator, a claim his own lawyers take issue with.
The jury has only two options: death or life in prison.
Last week, jurors.
On The Early Show Thursday, Abraham Scott, whose wife, Janice, died in the plane that hit the Pentagon,
"I want him to be put to death so that he can just be taken away from this world," Scott told co-anchor Julie Chen. "… I believe in the death penalty. And, after observing him in the courtroom and interacting with a lot of the family members, as well as talking to them about their loved ones, I want him to be put to death."
But Alice Hoagland, who lost her son, Mark Bingham, on Flight 93, which crashed in Pennsylvania, will testify that she wants Moussaoui's life to be spared.
"I am very sympathetic with Abraham Scott," Hoagland said to Chen. "… I understand and I respect every one of the opinions that are going to be expressed by the … families who lost loved ones on September 11th.
"I believe that Zacarias Moussaoui should be allowed to live for two reasons. Not because he has comported himself well. Not because he has not demonstrated his hatred for us in America. Not because he wouldn't, given the opportunity, kill us by the thousands on our own soil … but because, if he is executed he will be seen as a martyr in the eyes of his twisted fellows who compose al Qaeda. I don't want him to become a martyr.
"Secondly, and perhaps more important, we in America can now demonstrate that we are a nation of mercy, as well as a nation of laws and justice. We've already demonstrated our justice by allowing him to condemn himself out of his own mouth. And if we can transcend the kind of hatred he has shown toward us by allowing him to live, we will have given him a higher standard by which to live.
"I have always favored the death penalty. I've always cried with the mothers of children murdered. … This decision I've reached (about Moussaoui) is something that's come about in the last few weeks."
Scott observed: "I do believe that Jan (Janie Scott, his late wife) would want him to be put to death. We would have been married for 24 years on the 27th of December of 2001. And, I really miss my wife. I can understand (Hoagland's) opinion. I respect her opinion.
"However, I have two daughters right now, a 28-year-old, and a 20-year-old, and at the time of my wife's murder, they were in their early teens and early 20s, and it's something that I had to take over the responsibilities of my wife. I became dual-headed. And, if she was living now, we would have been sharing that responsibility together. It has been very trying to me. And the bottom line is … if it was the reverse case, she would have wanted Zacharias Moussaoui to be put to death."