In the event that there shall be no banking corporation, association, trust company or individual banker in the territory within which the depositary or depositaries of any public fund must, under the applicable laws of this state, be located to become eligible for selection, or in the event that the selected depositary or depositaries within such territory shall fail to accept such award or awards of such public funds as may be made, then the authority or authorities which are by law empowered to make such selection of depositaries and awards of public funds thereto are authorized and empowered to select as depositary or depositaries such banking institutions located outside the territorial limits aforesaid as such authority or authorities may deem the safest and most convenient depositary or depositaries for such public fund.