Benzema could be available for around the £35 million mark, but he has reportedly made it clear that he does not want to go to Anfield.
The refusal has left Rodgers to review his options, with Jay Rodriguez of Southampton the latest target.
Alex Crook of the Daily Star says Liverpool are set to raid the south coast club for a fourth time this transfer window as they look to fulfil the player's wishes to play in the Champions League.
However, Rodriguez is still out of action after a serious knee injury and is not likely to return until October, per Crook.
Benzema and Rodriguez are poles apart in terms of quality, and Liverpool fans must surely be concerned that world-class talent, such as Benzema, is not arriving after the exit of Luis Suarez.
Rodriguez would only be an auxiliary striker for the Reds if he signs, but Rickie Lambert has been purchased to fill this position and role.
Benzema is the type of predator Liverpool need to partner Daniel Sturridge in the attack. The Frenchman hit 22 goals in 46 appearances for Los Blancos last year, per WhoScored.com, and has the kind of reputation that will scare Liverpool's opponents in the next 12 months.
England international Rodriguez also had a fine campaign last year before being cut down by injury, ruining his chances of making the World Cup. He posted impressive stats for the season—hitting 15 goals in 30 Premier League starts—per WhoScored.com.
But it is questionable whether Rodriguez is the kind of player that Liverpool fans will want to see being purchased at this time.
However, the Mirror's Darren Lewis believes Remy may still end up being the striker that Liverpool sign and says QPR are unhappy with the collapse of the deal on medical grounds.
Rodgers appears to be patient in his pursuit of suitable targets and has said he is happy to wait until January to sign more players. He said, per Chris McKenna of the Express:
There are certain different areas where I want to bring in players, but they have to be the right type. There is money to spend, no question, but I won't spend it for the sake of it. It has to be the right type.
If that means I have to wait until January then that's what I will do. If the right ones become available we would like to do something because I still have a lot to do on that front—but if they're not available, I will wait.
Rodgers may be patient, but football has a way of punishing teams if progress slows.
The loss of Suarez is an obvious blow for Liverpool but the replacements brought in must be immediate and correct.
The transfer of Gareth Bale from Tottenham to Real Madrid is applicable here—Spurs brought in a host of good players trying to replace one world-class talent—and the team collapsed into mediocrity.
Liverpool could suffer the same fate as Andre Villas-Boas' side did last term. Suarez is not replaceable in respect of direct qualities, but swelling squad numbers will not help the Anfield club match last year's achievements.
Rodgers must bring in a recognised Champions League talent to lead his club forward, or regression might come swiftly to his team with the added strain of European football this season.