The Egypt international took his tally to 40 for the campaign against Bournemouth at the weekend and is only seven short of equalling Rush, who set the mark in 1983-84.
With four Premier League matches and a two-legged Champions League semi-final to come, Salah is eager to achieve another big milestone at Anfield, having already made the most prolific debut for the club.
"There's a big chance to break the record, to be number one for a club like Liverpool, it's something huge," the former Roma attacker is quoted by ESPN as saying. "I will be very happy if I break it. I'm happy about 40, I want to keep scoring, I want to be number one for the record.
"It's always good when you see your name with legends of a club like Liverpool, it's a different feeling and I'm very happy about that. But I always try to look at myself and I want more. I have 40 now, I'm telling myself, 'Why not? You can be number one!'
"I will try my best to break the record but, as I've said before, we play as 11 players, I play for the team. It's not about individual awards, it's something for the team. I'm sure if I have a chance to give it to someone else to score, I will - it's not about me, it's about us."
Liverpool have Premier League fixtures against West Bromwich Albion, Stoke City, Chelsea and Brighton & Hove Albion to come, plus two Champions League games against Roma with a place in the final at stake.