He was in the USA accompanying other members of his First Justice party for meetings with foreign dignitaries attending the United Nations General Assembly.
"At the moment he was going to be transported to court, while he was in the SEBIN waiting room, he jumped from the window of the building and fell, causing his death", Reverol wrote in a post on Twitter.
United Nations human rights spokeswoman Ravina Shamdasani told a Geneva news briefing her office was concerned by the news of Alban's death and because he had not been taken before a judge within 48 hours of his arrest as required under Venezuelan law. He said Alban told him that he had been interrogated but not tortured.
In a statement, the party said their member "was murdered at the hands of the regime of Nicolas Maduro".
The Venezuelan government and the opposition have also disputed the reasons behind Alban's arrest.
The official suicide claim drew swift denials from opposition leaders close to Fernando Alban and dozens of his supporters gathered outside the office of Venezuela's intelligence service yelling "Maduro killer!" contending that he had been murdered.
Venezuela accuses its neighbor Colombia of shielding the authors of the alleged assassination attempt.
The opposition claims that more than 100 Venezuelans opposed to Maduro are being held as "political prisoners", some for more than four years, with little access to the outside world and their legal rights routinely trampled on.
The government has denied that it is detaining people as political prisoners and said they were fairly jailed.
Municipal lawmaker Fernando Alban, 56, was jailed last Friday for alleged involvement in the explosion of two drones during a military parade in August that was led by President Nicolas Maduro, Interior Minister Nestor Reverol said.
He "requested to go to the bathroom, and when there he threw himself off the 10th floor", Attorney General Tareq Saad told Venezuelan state TV.
The chairman of the US Senate foreign-relations committee, Senator Bob Corker, called on Mr Maduro to explain the reasons for Mr Alban's death.
"Those of us who knew Fernando know that he could NEVER have acted against his life", he said on Twitter.
One of Alban's lawyers, Joel Garcia, told reporters that an autopsy conducted at the national morgue showed signs of severe trauma to the skull, chest, and hips as well as indications of a fall.
He said he met with Alban the night before in the tribunal and his client had seemed calm.
Also expressing concern was Venezuela's Catholic bishops' conference and Luis Almagro, the head of the Organization of American States and a sharp critic of Venezuela's socialist government, who called Alban's death "the direct responsibility of a torturing and homicidal regime".