If you ever wanted to start a rumor about a friend in Bitlife for whatever reason, it is possible. In short, you can start a rumor about someone via the option "start a rumor" under the relationship tab. But you will first need a friend in order to use this option.
You can do this really easily by giving someone a lot of gifts and performing nice actions on them. Eventually, the person will become your friend if your relationship tab is high enough. In other cases, you will get a pop-up of someone wanting to be your friend.
That said, once you got a friend made, you will be able to start rumors about them using the rumor option. However, be sure you want to use it beforehand, as it will most likely tank your relationship with that friend.
On the other hand, it is a requirement for this week's challenge called the Real Housewives challenge, so using the steps laid out here will help you complete that objective.
We also have a guide linked above that will go over the other objectives you will need to complete for the Real Housewives challenge if you are interested.