Chelsea have already publicly announced the capture of the Egypt international, but the move is still subject to him agreeing personal terms and passing a medical.
And Rodgers insists that his club will not give up on Salah until his move to Stamford Bridge was 100 per cent complete.
"The window is not closed yet so we haven't given up on him, no," Rodgers told BBC Sport after their FA Cup win over Bournemouth.
"Salah was a player we're interested in and we hoped we could do a deal but as of yet there is nothing done on it.
"The club is still looking for players we can bring in to strengthen the squad.
"It is a difficult market in January so we'll see what happens. If nothing comes in then the players I have have been outstanding for me this season and we'll enter into the final part of it with confidence if we have the same group of players."
Salah scored in three of the Swiss side's four recent meetings with Chelsea - a goal in each of this season's Group E Champions League wins, and one in the second leg of last term's Europa League semi-final.
But some have questioned whether or not the man dubbed 'The Egyptian Messi' would even be given a chance in the Blues first team, and it remains possible that the 21-year-old could be swayed by a promise of first team football.
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