The veteran broadcaster, best known for presenting Radio 2's breakfast show, said that he was on vacation when the parcel was delivered.
Wogan told Kirsty Young on Desert Island Discs: "My producer carried this parcel to the post room and was roundly castigated by the BBC - nearly lost his job for trying to blow himself up.
"Whoever sent in the bomb with my name on it couldn't have been much of a fan because I was on holiday."
The 73-year-old added: "It was very difficult. I was very conscious, for instance, that you'd [have to] come up [with] a cheery morning voice after some horrific bomb incident."
> Terry Wogan laments BBC compliance rules