I heard some where but I don't remember where that Vivian and Bill were offered a spin off of I Love Lucy but Vivian turned down the offer because she dislike Bill.

Yeah, that's true. Neither could stand to work with the other, so there was no way either of them were going to work together without Lucy and Desi there.

In addition to what dawsongirl said, I also heard that it was basically the end of any sort of "friendship" between Vivian Vance and Bill Frawley. They already disliked one another to begin with but the story goes that when Bill heard that Vivian turned down the series (my understanding is it wasn't even pitched to him before it got turned down by Vivian, thereby ending discussiions) he never spoke to her again, save for any notes, lines, etc. they might have to exchange during rehearsal or the actual show. A spin off could have made him a very rich man.