Brown was killed two days after Amber Guyger, a former officer with the force, was jailed following a trial in which he testified.
Civil rights campaigners have demanded an independent investigation into his death, which they called “deeply alarming and highly suspicious”.
But in a press conference on Tuesday, Dallas’s assistant police chief Avery Moore said rumours that Brown’s death was linked to Guyger’s trial were “false” and “have jeopardised the integrity of the city”.
Mr Brown named the men suspected of killing Brown as Jacquerious Mitchell, 20, Michael Mitchell, 32, and Thaddeous Green, 22, from Alexandria in Louisiana. The Mitchells, who are uncle and nephew, have both been arrested but Mr Green has not been caught.
Police said Jacquerious had told officers Brown shot him in the chest after a fight broke out during the drug deal. Mr Green is then said to have shot Brown twice and fled with his backpack and gun.
Jacquerious remains in hospital in a critical condition, Mr Moore told the press conference.
Police said they had confiscated 5.4kg of cannabis, 149g of THC cartridges and more than $4,000 (£3,300) in cash during a search of Brown’s home.
It is unclear how the three suspects came into contact with Brown or why they would have driven more than 300 miles from central Louisiana to purchase cannabis in Texas.
The police press conference came amid rampant speculation over Brown’s death. His shooting followed a jury sentencing Guyger, who is white, to 10 years in prison for killing her black neighbour in a case that sparked fierce debate over race, politics and policing.
Guyger told her trial she had shot Botham Jean in his fourth-floor flat in September 2018 after mistaking it for her own in the floor before. She was arrested on suspicion of manslaughter, a charge critics said was too lenient, but a grand jury later decided on the more serious offence of murder.
Brown, who was black and lived on the same floor as Jean, was one of several neighbours called by prosecutors to testify at the trial.
He told the court that on the night Jean was killed he heard what sounded like “two people meeting by surprise” and then two gunshots.
The NAACP Legal Defence and Educational Fund, which campaigns for racial justice, said Brown’s testimony had been “critical to officer Guyger’s conviction”.
“Now, the deeply alarming and highly suspicious murder of Joshua Brown increases the urgency of an immediate, independent investigation of every aspect of these two tragic killings,” it added.
Lee Merritt, a lawyer acting for the families of both Brown and Jean, said he had “no reason to believe [Dallas police’s] conclusions so far in the investigation are unreliable” but “some members of the community will have a difficult time accepting it” because of the force’s association with the Guyger trial.
He urged the police department to hand the investigation into Brown’s death to another law enforcement agency as a way to bolster trust.
But Mr Moore said it was reckless for people to speculate about the circumstances surrounding Brown’s death, adding it undermined the public’s faith in the department.
He told the press conference: “As you know, there’s been speculation and rumours that have been shared by community leaders claiming that Mr Brown’s death was related to the Amber Guyger trial and somehow the Dallas Police Department was responsible.
“I assure you that is simply not true and I encourage those leaders to be mindful of their actions moving forward because their words have jeopardised the integrity of the city of Dallas as well as the Dallas Police Department.”
A Dallas pastor who has pushed for police reform in the city said speculation around Brown’s death was fuelled by “generations” of mistrust for the police in the wake of wrongful convictions and abuse.
Reverend Michael Waters, of Joy Tabernacle African Methodist Episcopal Church, added: “There is cause for the community to be sceptical.
“There is a lack of credibility that the Dallas Police Department currently has and, frankly, it’s not just among brown and black communities.”