A federal judge last month refused the Pentagon’s motion to dismiss much of Amazon’s case. A few days later, Deputy Defense Secretary

Kathleen Hicks

said the department would review the project.

“We’re going to have to assess where we are with regard to the ongoing litigation around JEDI and determine what the best path forward is for the department,” Ms. Hicks said at an April 30 security conference organized by the nonprofit Aspen Institute.

Her comments followed a Pentagon report to Congress, released before the latest court ruling, that said another Amazon win in court could significantly draw out the timeline for the program’s implementation.

“The prospect of such a lengthy litigation process might bring the future of the JEDI Cloud procurement into question,” the Jan. 28 report said.

Ms. Hicks and other Pentagon officials say there is a pressing need to implement a cloud program that serves most of its branches and departments. The JEDI contract, valued at up to $10 billion over 10 years, aims to allow the Pentagon to consolidate its current patchwork of data systems, give defense personnel better access to real-time information and put the Defense Department on a stronger footing to develop artificial-intelligence capabilities that are seen as vital in the future.

Some lawmakers and government-contracting experts say JEDI should be scuttled because its single-vendor, winner-take-all approach is inappropriate and outmoded for mammoth enterprises like the Department of Defense.

These people say the Pentagon should move to an increasingly popular approach to enterprise cloud-computing that includes multiple companies as participants. Spreading out the work also reduces the risk of legal challenges from excluded companies, they say.

Rep. Steve Womack

(R., Ark.) called on the Pentagon last week to start fresh with a new contract-bidding process that would “enable best-in-class capability by prioritizing the ongoing competition that a cloud environment can promote.”

Should the Pentagon scuttle JEDI, the government could seek to patch together a new cloud program by expanding several existing Defense Department information-technology contracts, said

John Weiler,

a longtime JEDI critic who is executive director of the IT Acquisition Advisory Council, a public-private consortium that advises government and industry on tech-procurement best practices.

Microsoft has acknowledged the problems created by the delays but said it is ready to continue the project.

“We agree with the U.S. [government] that prolonged litigation is harmful and has delayed getting this technology to our military service members who need it,” the company said. “We stand ready to support the Defense Department to deliver on JEDI and other mission critical DoD projects.”

Amazon declined to comment for this article. The company has contended in court that then-President

Donald Trump

exerted improper pressure on the Pentagon to keep the contract from going to Amazon because it is led by

Jeff Bezos.

Mr. Trump has blamed Mr. Bezos for what he viewed as unfavorable coverage of his administration in the Washington Post, which Mr. Bezos bought in 2013 for $250 million. The Post says its editorial decisions are independent.

At the time, the Trump White House referred questions to the Pentagon, which denied that Mr. Trump or administration officials had any impact on the selection process.

Before the latest court fight,

Oracle Corp.

—one of the original bidders—had sued to halt the contract awarding process. Its 2019 lawsuit claimed that an Amazon employee who worked for the Pentagon in 2016 and 2017 helped steer the procurement process to favor Amazon, which then hired him back.

A judge subsequently rejected those allegations, allowing the bidding process to move forward.

Amazon has maintained that it got no favorable treatment from the Pentagon at any point, but the issue resurfaced last week, with

Sen. Mike Lee

(R., Utah) and

Rep. Ken Buck

(R., Colo.) sending a letter requesting a Justice Department investigation into alleged conflicts by that employee and others.

Last month,

Sen. Chuck Grassley

(R., Iowa) wrote a letter to Pentagon officials raising concerns about the agency’s oversight of the project and seeking more details about alleged conflicts of interest and possible improprieties, which some critics and rival companies say might have skewed the initial procurement steps in Amazon’s favor.

Several of the concerns raised in both letters had been reviewed previously. A federal judge in 2019 concluded that the former Amazon employee “did not taint” the program.

A Pentagon inspector general report last year determined that the Pentagon adviser didn’t violate any ethical obligations or give preferential treatment to Amazon.

Steven Schooner,

a George Washington University law professor who specializes in government contracting, said early questions about the Pentagon’s underlying procurement strategy for JEDI have grown over time.

“And all of that is before this case became one of the most jaw-dropping, head-scratching collections of conflicts of interest imaginable,” he said.

Write to John D. McKinnon at john.mckinnon@wsj.com