The Reds, who have already signed striker Daniel Sturridge this month, have been heavily linked with Inter Milan playmaker Philippe Coutinho but no deal has yet materialised.
Wales captain Ashley Williams, of Rodgers' former club Swansea, has also been mentioned as a potential target while Blackpool winger Tom Ince has long been the subject of rumours suggesting he could return to Anfield.
Rodgers said: "We hope to do some business between now and the end of the month.
"The owners have been brilliant in supporting (me), we knew we were maybe a little bit short in the summer.
"We all agreed this was a month we would try to reinforce, if we could.
"Very quickly we got Daniel Sturridge in and he has made a big impact, proving the owners were right to fund the money for him.
"With respect, £12million for a player like Daniel Sturridge is a real bargain and I think he will prove that over the next couple of seasons.
"We know we still need one or two additions more and the club are doing everything they can to assist me in that.
"If we can then great, if not we will have review it again in the summer."