To "float" an idea means to put it out in a tentative way. If I say "The moon is made of green cheese," I am stating a ridiculous idea, and if nobody believes it, I look ridiculous and lose credibility. But if I say "I hear a lot of people are saying the moon might be made of green cheese," I am just "floating" the idea. If nobody believes it, I lose nothing. I just stop saying it. If pressed, I can say "Oh, I don't believe that silly idea myself, I just heard that some people believed it." On the other hand, if everyone says "Yeah, right!" and all the talk show hosts start praising me, I can move on to just say "The moon is made of green cheese" and decide that this is a winning campaign issue for me and that I should keep talking about it. I just found a headline, "New Zealand's prime minister floats 'four-day week' as a way to help the economy." The story goes on to quote her: "I've had lots of people suggesting we should have a four-day week." She then gives reasons why it might be a good thing. She isn't saying they will do it. She hasn't even said she thinks they should do it. She is just floating the idea.

To suggest or present an idea, particularly an unproven one. From an idiom "see if it floats" which means to try something that you're not sure will work. It tends to be used like this to indicate that the "floater" is lying, misinformed, or simply weird, but not always. (Note that it's an unusual wording for English, not something you're likely to see in everyday writing.)