The female choir released their debut record In My Dreams on Monday (March 5), following the chart success of their first single 'Wherever You Are' back in December.
In My Dreams is currently trailing Springsteen's new album Wrecking Ball in the midweek chart, but analysts predict strong sales for The Military Wives in the run-up to Mother's Day on March 18.
Official Charts Company managing director Martin Talbot said: "The Military Wives Christmas campaign made for one of the most exciting festive chart races of all time – and this week's official albums chart race looks set to be almost as gripping.
"When the choir first came together last summer, I bet none of them believed they would be chomping at The Boss's heels in the charts within the year. If it were a movie script, you wouldn't believe it."
The final official chart positions will be released on Sunday (March 11).
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Listen to The Military Wives' new song 'In My Dreams', which features X Factor's Jonjo Kerr, below: