Transfers are not always possible. There are 2 transfer windows each season, during which players can be put on the transfer list. Outside the transfer windows you can't do this, and therefore also can't request a player to be put on the list.
Once a player is put on the transfer list, bidding can always be completed, even if there are multiple bidding rounds.
There is one transfer window between the end of the season and the start of the next. This is the longest window and the only one in which players can be set available for loan. The second, shorter, transfer window is roughly halfway during the season.
You can find out when the next transfer window starts (or when the current one ends) when you go to "transferlist" in your scout's office. If you don't have one, you can also go to the transfer page of any player and find the information there.