It was reported that the Saudi club would attempt to make a higher offer for Salah on their transfer deadline day but that failed to materialise.
Throughout the transfer window, Liverpool have been adamant that Salah would not be allowed to leave Anfield after former players Jordan Henderson, Sadio Mané, Fabinho and Roberto Firmino completed deals to Saudi clubs this summer.
"He didn't tell me [if he wants to stay] but he didn't have to," Klopp told reporters after the game. "He speaks with his training and performances and behaviour. We had meetings this week, and the meetings were not about what we did in the past, it was about what we will do in the future.
"Mo was with the players' [leadership] committee and had his moments where he was talking and it was nothing like 'by the way, this is only until next week' or whatever."
Salah's future could become a talking point once again when the transfer window reopens in January.
He is likely to return to action for Liverpool when they face Wolves following the international break.
Despite anticipation that Salah could be the latest superstar signing, the Saudi window ended relatively quietly. Demarai Gray joined Al Ettifaq from Everton and Luiz Felipe completed a move from Real Betis to Al Ittihad.
The transfer business of Saudi clubs has been spectacular enough, even without a deadline day deal for Salah, with a total outlay of $1 billion during the summer window.
The Associated Press contributed to this report.