German newsmagazine Stern said in its online edition that the video was shot in July 2006 in army barracks in the northern German city of Rendsburg. Excerpts were also shown on German television.
A barracks spokesman confirmed the existence of the video and said the instructor seen in it had been posted elsewhere.
The video shows the instructor first telling recruits to imagine they were fighting plane hijackers. At the end of the exercise he congratulates them, saying "That's good, the terrorists are dead."
He then tells them: "Now you're in the (New York City borough) Bronx. A black truck stops in front of you. Three African-Americans get out and insult your mother grossly. Every time before you pull the trigger I want to hear you shout 'mother......' very loudly."
The junior soldier is then shown executing the order, shouting the required word before each shot, encouraged by his superior.
New York leaders react
Bronx Borough President Adolfo Carrion Jr. said that he was saddened by the incident, which showed that German military officials had to do more to educate soldiers.
"Clearly these folks don't know anything about African-Americans or the Bronx," he said, adding that he expected an apology from the Bundeswehr.
Carrion, who recently visited Germany to promote the Bronx, said that he was willing to sit down with Bundeswehr officials to talk about the case.
New York Governor Eliot Spitzer said he trusted the "deeply offensive comment" did not reflect training practices of the German army.
The Rev. Al Sharpton, a prominent African-American rights activist, meanwhile called on US President George W. Bush to condemn the video.
In Bonn, a Bundeswehr spokesman said the army had launched an internal inquiry into the incident, possibly leading to disciplinary measures against two instructors. The ministry said its investigation would last two to four weeks.
The Defense Ministry said the instructor's statements were "absolutely unacceptable," according to spokesman Thomas Raabe, who added that it "contradicts the training standards of the German army."
The army has recently had to deal with a number of scandals.
In the northwestern city of Münster, 18 German soldiers are on trial for alleged mistreatment of 163 recruits with electrical shocks.
Last year, German soldiers were photographed in Afghanistan, posing with human remains. An inquiry involving 20 soldiers was launched, and six have already been suspended from duty.