It was expected the Saudi club would make a world record bid in the region of £215m for the 31-year-old forward, but no such offer was confirmed.
The Reds had rejected a £150m move for the player on Deadline Day and were bracing themselves for an improved offer on Thursday.
It is being reported that no such offer arrived ahead of the Saudi transfer window closing at 10pm (UK).
The Egyptian will now stay on Merseyside with his current contract due to expire at the end of the 2024/25 season.
It has been an unprecedented transfer window for Saudi Arabia's Public Investment Fund with its clubs spending £700m for stars including former PSG forward Neymar, Real Madrid's Karim Benzema and Bayern Munich's Sadio Mane.
They join ex-Premier League internationals Riyad Mahrez, Roberto Firmino, Jordan Henderson, N'Golo Kante and Fabinho in the Middle East, with five clubs - Al-Ahli, Al-Ettifaq, Al-Hilal, Al-Ittihad and Al-Nassr - buying the most players.
Salah has been integral for Jurgen Klopp's team since he joined from Roma in 2017, scoring 188 goals in just 209 games.
It is likely Al-Ittihad will test Liverpool's resolve with a renewed bid for the attacker when the window re-opens in the new year.
