When it comes to stars as big as Ayumi it's very likely she could be offered positions in various TV shows, but it's even more likely that random rumours starts floating around.
Rumours are rarely entirely untrue though, she probably was offered something and then people started talking about how she was scheduled as a hostess for a lot of shows
.
Unless someone can find an old article stating which exact shows those was, I belive we will never know the entire truth.