Diego Simeone's side exited the competition on away goals after drawing 3-3 at the Wanda Metropolitano in the second leg of their last-16 tie.
Saul had to be replaced 11 minutes into the second half and Atleti said a subsequent injury assessment has determined the midfielder has suffered a "high-grade post-traumatic myofascial injury to his thigh".
A club statement read: "Saul Niguez has undergone this Thursday [an] MRI scan at the Clinica Universidad de Navarra to know the exact extent of the injury he suffered in the Copa del Rey match against Girona and that prevented him from finishing the match caused by a thigh bruise, being replaced by Rodrigo.
"Our midfielder suffers a post-traumatic myofascial injury to his thigh. He is pending clinical evolution."
Saul, who began his youth career with Real Madrid and labelled a possible return there as 'unthinkable' , has worked his way up the ranks of Atletico originally playing for their B-side before making his first-team debut at just 17.
The 24-year-old is no stranger to injury struggles either, revealing in early 2017 he was urinating blood after every match having seriously hurt his kidney in 2015.
"For the past two seasons I've been using a catheter and after every match or training session I would pee blood and it was certainly not an easy time for me," Saul told Mega at the time.
"I was gambling with my health and driven by the desire to play first team football for Atleti."
Saul's resilience was rewarded later that year as he was handed a five-year contract extension that keeps him tied to the club until 2026.