Jeff Bezos ’ space company has escalated its battle to build a moon lander for the National Aeronautics and Space Administration, suing the federal government over the agency’s decision to award a sole contract for that work to Elon Musk’s SpaceX.

An affiliate of Blue Origin LLC on Friday sued the government in the U.S. Court of Federal Claims over the lander, court records show. In a related filing, the company said the suit protests NASA’s decision on the lander, which it called improper. Blue Origin said it filed the suit under seal to protect sensitive information.

Last April, NASA awarded Space Exploration Technologies Corp., the formal name for SpaceX, a $2.9 billion contract to develop a vehicle that it would use to land astronauts on the moon under its Artemis program. Blue Origin and Dynetics, a unit of Leidos Holdings Inc., had also competed to create a moon lander for the agency but failed to win part of the contract.

On Monday, a Blue Origin spokesman said the company filed the suit to remedy what it said were flaws in how NASA handled awarding the contract for the moon lander. The Verge earlier reported on the lawsuit.

NASA officials are reviewing the case, a spokeswoman said.

The agency has said in the past it chose SpaceX in part because of budget constraints and that it was permitted to only choose a single provider for the vehicle. The agency has also said its decision to award SpaceX the contract was a first step, and not the final one, in its plans to hire companies to provide moon-landing services.

Mr. Bezos has sought to convince NASA to permit Blue Origin to participate in the moon-lander program. Late last month, he offered to waive up to $2 billion in fees from its bid and provide other benefits in a letter sent to agency administrator Bill Nelson.

The company had also lodged a protest with the U.S. Government Accountability Office over NASA’s decision, but the GAO said the agency didn’t violate any procurement laws or rules in choosing to only award SpaceX the contract.

SpaceX didn’t respond to a request for comment.

Blue Origin has pressed its case for the lander outside of court as well. On its website, the company posted a graphic that criticized the SpaceX lunar lander, implying that its exit hatch is too high above the moon surface.

Such criticism drew a response from Mr. Musk, who said in one tweet if “lobbying & lawyers” could get you into orbit, Mr. Bezos would have reached Pluto. A Blue Origin spokesman didn’t respond to a request for comment.

Amazon founder and former CEO Jeff Bezos successfully completed the first manned mission of Blue Origin’s New Shepard spacecraft. In this video, WSJ looks back at the remarkable career of the world’s richest man. Photo: Joe Raedle/Getty Images The Wall Street Journal Interactive Edition

Blue Origin’s partners for its lander include Lockheed Martin Corp. and Northrop Grumman Corp.

Write to Micah Maidenberg at micah.maidenberg@wsj.com