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."
Rodgers, speaking at a press conference to preview this weekend's FA Cup fourth-round tie at Oldham, would not give any names.
He said: "It is all speculation and I would only comment once we get a player in the door."
Liverpool were frustrated during the summer transfer window, most notably as they allowed striker Andy Carroll to join West Ham on loan and then failed to secure a replacement.
The Merseysiders were thought to be keen to sign Clint Dempsey from Fulham but a potential deal collapsed on deadline day and the American joined Tottenham instead.
Rodgers, who only took charge last summer, believes the situation at the club is different now.
He said: "It was difficult all-round in the summer. A new manager with no recruitment staff, the lines of communication - it was difficult. We were so busy in pre-season.
"But I have got to say from that moment on - we got the recruitment team in, the lines are very clear and everyone has an understanding of where we are at. It is working how we envisaged it to work.
"There is a small group who'll look at the players we want to bring in and that relationship will only get better as go on. We're all happy and comfortable."
Rodgers is likely to make changes for the trip to Boundary Park on Sunday as he keeps an eye on the midweek game at Arsenal and next weekend's visit to Manchester City.
Yet Rodgers will not be taking the tie lightly as he recognises how important it is to bring silverware to the club.
He said: "We want to do well in the cup. This is a club that has got a real proud tradition in this competition.
"We want to go as far as we possibly can, without taking away from the two very important league games coming up.
"It is a chance for me to look at the squad and games like this, the squad really comes to the fore.
"I have great trust in the players, they been absolutely brilliant in my time here.
"I know whichever team I put out, Sunday, Wednesday and the following week - they will be strong teams able to get a result.
"But we'll be taking a strong squad because we don't want to hold back. It is a tournament we want to do well in."
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