The photo posted by the White House on its official Instagram account had President Biden with a member of the Army's elite counter-terrorism unit.
However, in the same photo, the faces of three other Delta Force fighters were also visible in the image.
A distinctive tattoo on the soldier's right arm was clearly recognisable.
After reading the blunder, the White House deleted the photo an hour later from their social media account.
A White House spokesperson expressed regret over the error and any resulting complications. The faces and identities of special operators are typically kept confidential to safeguard against enemy identification in the event of capture.
Pictures of Delta Force members, especially ones revealing their faces, are exceedingly rare.
President Biden dispatched special operators to Israel to provide comprehensive assistance in the hostage crisis, encompassing intelligence sharing and the deployment of experts from various US government agencies. The Israeli government has confirmed that 203 hostages are currently held by Hamas following the group's attack on southern Israel on October 7.