Sturridge had been summoned to appear at a court in Los Angeles over a dispute over the reward, but the former Liverpool and England striker did not appear, resulting in the warrant being issued.
Foster Washington, also known by his rap name Killa Fame, claims he is the person who returned Sturridge's dog, Lucci, after it was stolen four years ago.
Sturridge had publicly offered a $30,000 reward and Washington is now pursuing legal action as he says he did not receive the payment.
The ex-Chelsea attacker has since taken to social media to declare his innocence, saying that he already paid the reward to the person who brought his dog back and that Washington does not deserve any money from him.
"The news yesterday was the first time my family and I heard that someone is continuing to try to get money from me, following the theft of my family's dog, Lucci, which happened four years ago," he wrote in an Instagram Story.
"The truth is, I personally paid a reward to the young boy who found him. He was delighted with the reward, as were my family and I to get Lucci home. The person seeking payment is not due any money, I already paid the young boy who found Lucci.
“I was unaware of the court hearing even taking place. The legal paperwork was sent to the AirBNB property which we had left within 24 hours of the burglary.
"Therefore, I have not had the opportunity to properly represent myself in relation to these false claims. With that being said, I have now instructed California lawyers to deal with matters on my behalf.
"I am confident that this will be resolved in my favour quickly and put an end to the speculative claim for good."