BRUSSELS—As NATO relocates its headquarters here, an Italian Renaissance oil painting is going elsewhere, a departure some alliance officials say points to a deeper problem with the organization’s mission and cohesion.

The biblical scene, painted by Orazio Fidani around 1650, has hung outside the council chamber of the North Atlantic Treaty Organization since 1988 as a loan from Italy, one of the alliance’s founding members. Italy’s delegation recently told planners the artwork will return to Florence rather than cross the street to NATO’s glistening new building, which the organization is now starting to occupy.

Some allied officials say they view the move as symbolic of the frustration with the alliance in Italy and other Southern European members, even though Italy’s ambassador to NATO insists the painting’s return to the Uffizi Gallery is routine and “not a metaphor.”

Now NATO officials say they have something to offer those allies. Planners have proposed creating units specialized in training foreign militaries and quickly sending them to help local authorities stabilize trouble spots in the Middle East and North Africa, including Tunisia and Libya.

“NATO has a sales job to do in Italy and other southern countries to show it is relevant,” said Stefano Stefanini, a former Italian ambassador to the alliance.