Amazon is the undisputed leader in cloud computing infrastructure. But that doesn't mean the company is close to winning a major cloud contract from the U.S. Department of Defense, contrary to at least one media report.

The deal, which is reportedly worth up to $10 billion, could make a material difference for a smaller cloud provider, such as Oracle or Google. In Amazon's case, the dollar amount isn't that big a deal -- Amazon Web Services books around $5 billion in revenue per quarter -- but it would give the company even more government credibility and lead to further deals.

The final request for proposal with updated terms, which tells companies what they'll have to deliver to get the contract, has not been released yet. It will become available in early May. The contract will be awarded in September, then companies can then file protests.

U.S. Navy Commander Patrick Evans, a Department of Defense spokesperson, reiterated that the Pentagon's process is "transparent" and will remain "a full and open competition."

"No companies were pre-selected. We have no favorites, and we want the best solution for the department," Evans told CNBC.

Chief Pentagon spokesperson Dana White also addressed speculation Thursday that Amazon was in the lead to take the lucrative defense contract.

"The secretary has been very clear that we need to be good stewards of the American people's money," White said. "So, nothing is taken for granted and nothing is presumed. We will get a full, open and transparent competition, and this is the first of many competitions with respect to the cloud."

In the midst of all this, President Trump has been expressing concerns about Amazon on Twitter, and on Monday Vanity Fair reported that his advisers were suggesting that he try to "cancel Amazon's pending contract" with the Pentagon.

A day later, Oracle CEO Safra Catz had dinner with President Trump, along with technology investor Peter Thiel, sources told CNBC.

"I think she and Peter were very surprised by how much the president actually knew about this. He knew a lot about it," a person familiar with the dinner conversation told CNBC.