Because of transverse gradients of refractive index, magnetospheric ducts which guide whistlers are not isolated. The whistlers are likely to leak from the ducts through a forbidden region owing to barrier penetration by evanescent waves. This phenomenon is investigated theoretically by using the WKBJ approximation and phase integral methods. A new type of mode called a quasi-trapped mode is discussed. It is shown that, if a wave packet is incident on a duct capable of sustaining quasi-trapped modes, certain components of its angular spectrum are selectively trapped. These travel in quasi-trapped modes for considerable distances before leaking from the duct. Expressions are derived for the distance travelled in terms of the width of the duct, the width of the barrier region, and the properties of the medium. A connexion between this type of solution and a solution in terms of leaky modes is discussed.