The Chelsea forward is expected to move to Anfield when the transfer window opens and had a medical on Merseyside on Sunday.
And it has been speculated that Sturridge may have a clause in his contract stipulating that he will always be played up front, having been consigned to the wing on occasion at Chelsea.
But Rodgers said at a Christmas Eve press conference: "You can keep writing and saying these things but no one will come into this club and demand where they play, that's for sure."
Asked if there was any more news to report on the prospective signing, Rodgers said: "Nothing will be complete until we've signed a player at the beginning of January.
"Once we've signed a player, I'll have no problem in speaking about who we've signed. But, until anything is done on paper, there is nothing to say.
"We'll always do our business privately and in the way we want to do it. As you've seen over the last number of weeks, there are always going to be things written about players, clubs and dealings.
"We're confident we can get some business done early in January, which will really help the squad. Until that happens, there is nothing really to add.
"We don't want to make many (additions) in January but the one or two we do, I think can make a difference and help us. If we can do that, I think we'll have a good second half to the season."
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