The offer came just months before the two sides met in the Champions League final in Kiev, with the forward injured by Sergio Ramos
The forward has been linked with a move away from Anfield during his goal-filled time at the club, with Madrid and Barcelona usually reported as the interested parties.
Ramzy has said the Spanish giants indeed made a “really good” offer to sign Salah, but that the forward had decided he was happy on Merseyside.
“I was talking to Salah and he told me that Real Madrid had sent him an offer.
“The offer was really good, but Salah discussed it with me and Mr Cuper, and he decided to stay with Liverpool because he was comfortable there.”
Ramzy wasn’t appointed assistant manager until Cuper was replaced by Javier Aguirre, but said he was at the training camp in order to support the squad.
The offer would have come just months before Salah and Liverpool faced Real Madrid in the Champions League final in Kiev.
Madrid won 3-1 after goals from Karim Benzema and Gareth Bale, but the match was overshadowed by an early injury to Salah, who went off in tears with a dislocated shoulder after a challenge from Sergio Ramos.