One of the officials was holding the rope which was connected to the sleeves as the man was accompanied along a street in Galveston.
Photos of the occurrence were posted on Twitter by Adrienne Bell, who is running for Congress in 2020.
She said: “It is hard to understand why these officers felt this young man required a leash, as he was handcuffed and walking between two mounted officers.
“It is a scene that has invoked anger, disgust and questions from the community.”
The city’s police department said Donald Neely, 43, had been arrested on suspicion of criminal trespass.
It said the two officers were “familiar” with Mr Neely and were “aware he had been warned about trespassing” at the location several times.
Mr Neely was reportedly led on the rope for about two blocks to “where the mounted patrol unit” was based.
Police chief Vernon L Hale said: “I must apologise to Mr Neely for this unnecessary embarrassment.”
The department said “a transportation unit was not immediately available at the time of the arrest and a man was handcuffed and escorted beside two police officers on horses”.
It added: “While this technique of using mounted horses to transport a person during an arrest is considered a best practice in certain scenarios, such as during crowd control, the practice was not the correct use for this instance.”
The department admitted “our officers showed poor judgement in this instance and could have waited for a transport unit” to arrive.
We have verified with law enforcement officials in Galveston, that the photograph taken in Galveston is real. It is hard to understand why these officers felt this young man required a leash, as he was handcuffed and walking between two mounted officers. pic.twitter.com/bEFZnn4qmH
— Adrienne Bell (@AdrBell) August 5, 2019
The officers did not show any malicious intent but the department was changing its policy to prevent the use of the technique in future, it continued.
Ms Bell said she applauded the police chief’s “swift action” in discontinuing the practice.
But she added that “questions about transparency, community policing in the community, and accountability still remain. There are also concerns about arrest procedures for persons who are known by the police department, as having mental issues”.
Ms Bell said she would soon invite law enforcement leaders to a town hall meeting so they could face the concerns of community members.