This hotly contested contract was awarded to Microsoft on Friday night over Amazon in a one-month battle.
According to Snodgrass's book, Trump called Mathis in the summer of 2018 and directed him to "pull Amazon" out of the possibility of bidding.
Task and Purpose received an expanded copy of the book. CNN has not yet seen the book.
For several years, Trump has expressed his dissatisfaction with Amazon and Jeff Bezos, who also owns The Washington Post. He accused Amazon of taking advantage of the postal service, although independent investigations disagreed. He also linked his adverse view of the Washington Post to Amazon, though the mail clearly states that it is managed separately.
"Sharing the story with us during the Small Group, Mathis said, 'We will not do this. This will be done from the book, both legally and ethically, "Snodgrass writes according to Task & Purpose.
The White House did not immediately respond to a request by CNN for comment.
In response to the book and some of the other allegations it contains, a spokeswoman for Matisse earlier this week released a statement saying that Snodgrass is a "junior employee" and "plays no role in decision-making". Snodgrass responded to these allegations by showing a quote he received in which he said he played a vital and influential role in the Pentagon's communications.
Amazon and Microsoft respond
In a statement, Amazon says he is surprised by the decision.
"(Amazon Web Services) is the clear leader in cloud computing and detailed assessment, whose purely comparable offer clearly leads to a different conclusion," the release said. "We remain deeply committed to continuing to innovate for the new digital battlefield, where security, efficiency, sustainability and scalability can be the difference between success and failure."
In a statement, Microsoft quoted reporters from the Department of Defense's message.
"We're working on it right now. See the DOD message for more information," the statement said.
In July, Trump promised that his administration would look "strongly" at the Pentagon's contract plan, which said "some of the largest companies in the world" had complained, including IBM, Oracle and Microsoft.
However, many independent reviews of the process have found little evidence of misconduct.
In a statement, the Department of Defense's Office of the Inspector General stated that investigators were nearing the completion of their work reviewing the award of the contract.
"To date, we have not found evidence that we believe would prevent the Home Office from deciding on the award of the contract," said Duren Allen, spokesman for the Inspector General's office. "We hope to have a finalized report on our findings by the end of November that we intend to publish to the fullest extent possible."