The Reds have agreed a £12m deal with Chelsea for England striker Sturridge and a £60,000 per week contract with the player.
Sturridge was due to undergo a medical at Liverpool on Sunday.
"Nothing will be complete until we've signed a player," said Rodgers.
Players to have joined Liverpool since Brendan Rodgers took charge
Fabio Borini from Roma £10.5m
Joe Allen from Swansea £15m
Oussama Assaidi from Heerenveen undisclosed
Samed Yesil from Bayer Leverkusen undisclosed
Speaking at a news conference on Monday, Rodgers added: "Once we do, I'll have no problem in speaking about who we've signed.
"But until anything is done on paper, there is nothing to say."
Chelsea's interim manager Rafael Benitez said after his side's  8-0 win over Aston Villa on Sunday that he believed Sturridge had travelled to Merseyside for a medical.
"The information I had was that he was there, passing the medical, but we don't have any more information," he added.
"Maybe he will come back to say hello, or goodbye. We will see. He's a good player. He has all the talent."
Reports had claimed that Sturridge wanted a clause in his contract ensuring that he would be played up front.
But Rodgers said to journalists: "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."
Former Manchester City striker Sturridge has made seven league appearances for Chelsea this season, scoring one goal.
His last league appearance for the Blues was in the 2-1 defeat by West Bromwich Albion on 17 November.
Sturridge has not featured since Rafael Benitez arrived at Stamford Bridge, due to a hamstring injury.
Rodgers is keen to add to his striking options after letting Andy Carroll join West Ham on loan in August and then missing out on former Fulham striker Clint Dempsey on deadline day.
"We'll always do our business privately and in the way we want to do it," added Rodgers.
"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 signings 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."
Liverpool manager Brendan Rodgers says he is confident he can strengthen his squad during the January transfer window, but refused to confirm that Daniel Sturridge is close to signing.