Amazon.com Inc and Microsoft Corp were selected in April to continue competing for the Pentagon cloud computing services that is part of a broad modernization of Pentagon information technology systems.
The selection left Oracle Corp and IBM Corp out of the competition for the contract for the Defense Department's Joint Enterprise Defense Infrastructure Cloud, or JEDI.
Trump has taken several swipes at Amazon since becoming president, complaining of unfair business practices and that the online retailer does not pay the U.S. Postal Service a fair rate for package delivery.
Amazon did not have an immediate comment.
Trump's criticism stems in part from his oft-expressed dislike of the Washington Post's critical coverage of his administration. Amazon Chief Executive Jeff Bezos privately owns the Washington Post.
Oracle has complained about the contract award process, including expressing concern about the role of a former Amazon employee who worked on the JEDI project until recusing himself, then later left the Defense Department and returned to Amazon Web Services.
Oracle earlier this month lost a lawsuit challenging the contract award. A judge ruled Oracle did not have standing to claim it was wronged by the decision because it did not meet the contract requirements.
Its chief executive, Safra Catz, who was a member of the executive committee of Trump's transition team after he was elected, told reporters in April that she has met with Trump to discuss the contract, telling him commercial customers often use more than one cloud.
(Reporting by Steve Holland; Writing by David Alexander; Editing by Susan Thomas and Jonathan Oatis)