Llorente scored two goals in the round-of-16 second-leg at Anfield to lead Atletico to a 4-2 aggregate victory over Liverpool. The 25-year-old had scored just once for the Spanish club before the game, with his brace handing Liverpool their first home defeat for 25 games in the European competition.
Llorente posted a picture of the dog via Instagram, naming his pet "Anfield" in the photo's caption.
The match, which went ahead despite reservations from a number of politicians, was the final game to be played by either side before their remaining games in all competitions this season were postponed due to the coronavirus pandemic.
Liverpool city council's director of public health Matthew Ashton said on Friday that the clash could be a factor in the rise of cases in the city.
The coronavirus outbreak has since seen football stopped around the world with the Champions League being suspended indefinitely along with the Europa League.
Atalanta's Champions League clash with Valencia in February has also been called into question, with intensive care director Luca Lorini saying that the match acted as an accelerator for cases in the Italy.