So, 4 out of 5 rumours are true?

I'd say make it like this:

1. Rumour that's hard to believe but true
2. Rumour that's true, but not too interesting
3. Rumour that's based on something that actually happened, but the facts have been distorted
4. Rumour that makes the player look good, but is false
5. Rumour that makes the player look bad, and is false

If there are only 2-3 other players, each one gets 1 or 2 rumours. If there are a lot of players, you might duplicate some of them. You could ask the player to pick the rumour that's most popular about them, and some extra copies of that one get handed out.

So, about half the rumours are true, or based on something true. Half are false or distortions. There's one that the character might want people to continue to believe, even if it's false. Another one that is a lie that the player can't seem to shake. Another one where it's a true story, but maybe the character is just tired of having to re-tell. On the other hand, maybe it's a tale of heroism that they love to tell people about.

IMO having one that makes the player look good, but is false, and one where the facts have been distorted are the most interesting ones. A paladin-type character would probably want to clear up the rumours even if they made that paladin look good. A rogue might want people to believe all the rumours even the ones that make them look bad, just so that nobody really learns the truth about them. A bard might want to string people along, building more rumours and making even the true stories into fanciful tales.