The matter is with the US Government Accountability Office (GAO), which is likely to announce its ruling by 16 February, 2016.
Expected to be inducted into the USAF in the 2020s, the LRS-B is a heavy-payload stealth aircraft capable of carrying thermonuclear weapons.
The USAF estimates that it requires 80-100 bombers, each costing around $550m.
In October, Northrop Grumman was awarded the contract, which could be worth up to $80bn. The company previously built the B-2 Spirit stealth bomber for the US.
The decision was, however, questioned by other contenders Boeing and Lockheed earlier this month.
Asking the US GAO to review the contract, the two companies termed the selection process as fundamentally wrong.
They said in their complaint: "The cost evaluation performed by the government did not properly reward the contractors’ proposals to break the upward-spiraling historical cost curves of defense acquisitions, or properly evaluate the relative or comparative risk of the competitors’ ability to perform, as required by the solicitation.
"That flawed evaluation led to the selection of Northrop Grumman over the industry-leading team of Boeing and Lockheed Martin, whose proposal offers the government and the warfighter the best possible LRS-B at a cost that uniquely defies the prohibitively expensive trends of the nation’s past defense acquisitions."
The USAF said that Northrop Grumman offered to build the bombers at $515m per unit, which is lower than the approved cost of $550m.
Reacting to the complaint, Northrop Grumman vice-president of strategic communications Randy Belote said: "Northrop Grumman offered an approach that is inherently more affordable and based on demonstrated performance and capabilities.
"As the only company to ever design and build a stealth bomber, we offered the best solution for our nation’s security.
"We look forward to the GAO reaffirming the Defense Department’s decision so we can continue work on this critically vital programme."