At Salzburg, Clark will be reunited with former Liverpool assistant manager Pep Lijnders, who took over at the club in May.
“After an incredible three years at Liverpool FC, I’ve decided it’s time for a new challenge,” Clark wrote on social media.
“I’ve learned so much during my time at the club and [I] am very thankful for the opportunities and experiences I’ve had. Thank you to all the fans who supported me on this journey.”
Liverpool began their Premier League campaign under new manager Arne Slot with a 2-0 win over Ipswich, and take on Brentford this Sunday.