He also said the mosquito-borne Zika virus, linked to birth defects, would not be a threat.
Rio de Janeiro's city government said earlier this week it had cancelled its contract with the company constructing the velodrome after the firm filed for bankruptcy protection.
"That is one of the points we explained to the IOC," Nuzman told reporters after the organisers made their final progress report to the International Olympic Committee's Executive Board.
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"The Rio mayor was there (via video link) to provide these answers. At the end of June it will be in condition for athletes to train there...the mayor gave these guarantees because he is in charge of the construction."
The indoor cycling test event scheduled for the end of April was cancelled in March because it was not ready but the IOC now says it will be staged on June 25-26.
The construction of the venue has now been handed over to Engetecnica, a Brazilian company that has been working on the project as a sub-contractor since February.
Scientific consensus
Nuzman also said that, according to the World Health Organisation (WHO), Zika does not present a health risk that would warrant moving the Olympics.
U.S. health officials have said Zika infections in pregnant women can cause microcephaly.
The WHO has explained there is strong scientific consensus the virus can also cause Guillain-Barre, a rare neurological syndrome that leads to temporary paralysis in adults.
Nuzman added several athletes, including world number one tennis player Novak Djokovic, Brazilian footballer Neymar and sprinter Usain Bolt had said they were not concerned about Zika.
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However, he also explained that no female athlete had yet responded in the same way as Djokovic, Neymar and Bolt, adding there was no sign the virus was keeping any athlete away from the Games.
Rio spokesman Mario Andrada said there was no need to stage any public information campaign because of what he said was a low infection rate during the winter months in Brazil.
"We had zero cases of Zika in 44 test events involving 7,000 athletes and 8,000 volunteers," Andrada said. "There is no reason to engage in a public campaign.
"We don't need to push and emphasise. Women planning to get pregnant have to take extra care and it is up to them and their families to decide."
The Olympics start on August 5.
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