The question can be taken more than one way. "The People Offer to Expand Your Palace" aka the F9 (?) screen is purely for personal amusement; there is no affect in the game.
After the Masonry tech you can build a palace in a city. If completed, that will move your palace to that city, but may people start building the palace as a prebuild. Example: A player want's to build The Hanging Gardens and expects to have Monarchy in 60 or so turns. To improve his chance of building it first he may start building the palace and then when he discoveres Monarchy change the palace build to Hanging Gardens and be well on the way to completing it.