After searching for a bit, I finally found the audio tone for the mobile phone used in The Big Lebowski! Presented here for your personal ringtone amusement:
Dude: Fuck it.
Donnie: Where you going, Dude?
Dude: Home, Donnie.
Donnie: Phone’s ringing, Dude.
Dude: Thank you, Donnie!
old 90s GSM ringtone