The Browns said Gordon and his attorney are "working with the Cuyahoga County Courts to resolve this matter."
The warrant was issued Aug. 31 after Gordon failed to respond to several subpoenas over the course of a year demanding he take a DNA test.
Gordon's attorneys, in a statement, said Gordon was unaware until Wednesday that he had been requested to be at Cuyahoga County Juvenile Court, and that he made himself available immediately once he became aware.
According to the attorneys, he attended court Wednesday, a hearing was held and the "matter was continued until November should a further hearing be needed."
The NFL did not immediately respond for comment. After a one-year suspension, Gordon was reinstated conditionally on July 25 by commissioner Roger Goodell, though he was also suspended for the first four games of the season.
An NFL spokesman said Wednesday that the warrant would not have any effect on Gordon's conditional reinstatement. Gordon was in the team facility on Wednesday, but because he is suspended, he is not permitted to talk to the media.
In July 2015, Christina Lockhart from Maple Heights, Ohio, filed a paternity suit stating Gordon was the father of her daughter, who is now 1 year old. The order sought child support.
Gordon was ordered to provide a DNA sample; he had until Sept. 25, 2015, to do so. The complaint stated that nobody other than Gordon was presumed to be the father. It also stated the woman is on welfare and needs child support from Gordon.
Subpoenas were sent to Gordon in Los Angeles and to several different addresses in Cleveland, but he never responded.
Gordon was due to appear at a hearing on Aug. 31, but he was not present, which led to the warrant.
The NFL suspended Gordon for the entire 2015 season for violations of the league's substance abuse policies. He also was suspended 10 games in 2014, and two games in 2013.
The team had said that Gordon's return to the team was going well.
"He still is doing well, in my opinion," Browns coach Hue Jackson said Wednesday. "I can't tell you about everything that goes on outside of this building -- but I know in this building, in our meetings with me, with our coaches, with the players, he has been outstanding. That's what I have to judge on.
"Now we'll find out about this other part and know a little bit more about it as we move forward. I have nothing but great things to say about Josh."
Jackson said he did not have many details about the arrest warrant, so for now, "All I can do is support and let's make sure we do the right thing, whatever that is pertaining to the situation. Hopefully this will be wrapped up sooner rather than later."
The Associated Press contributed to this report.