The Briton is believed to have been from Blackburn.
A statement attributed to the suspect’s brother Gulbar read: “It is with great great sadness I will confirm my brother Faisal passed away in Texas, USA this morning.
“We are absolutely devastated as a family. We can’t say much now as their is an ongoing FBI investigation.
“We would like to say that we as a family do not condone any of his actions and would like to sincerely apologise wholeheartedly to all the victims involved in the unfortunate incident.
“Sitting in the incident room all last night at Greenbank until the early hours liaising with Faisal, the negotiators, FBI. And although my brother was suffering from mental health issues we were confident that he would not harm the hostages.
“At around 3am the first person was released then an hour later he released the other three people through the fire door unharmed.”
The Foreign, Commonwealth and Development Office subsequently has confirmed the man’s nationality, with a spokesman saying: “We are aware of the death of a British man in Texas and are in contact with the local authorities.”
Local reporters said they heard the sound of explosions, possibly flashbangs, and the sound of gunfire shortly before Texas Governor Greg Abbott announced the crisis was over.
Mr Abbott tweeted: “Prayers answered. All hostages are out alive and safe.”
President Joe Biden, who was briefed on the crisis as it unfolded, praised the “courageous work” of state, local and federal law enforcement in freeing the hostages.