The ministry became aware of a picture from a regimental lunch after it was posted on social media.
It said it is looking into who was in attendance and whether there were any breaches.
The social media post reportedly showed people dressed as paedophile Jimmy Savile and a Zulu, according to the Sun newspaper.
An Army spokesman said the fancy dress competition is a tradition.
"We are aware of a social media post showing photos from a fancy dress competition at a regimental Christmas lunch," an Army spokesman told the PA news agency.
"While the event is a long-standing unit tradition, we are very clear all activities should be in line with the Army's high values and standards.
"If we find that these standards have been breached there are procedures for dealing with all forms of unacceptable behaviour."