The Irish defender is among the highest earners in MLS and is currently in discussions with another club about his future as Toronto seek to free up space on the wage bill
Nelsen explained that O'Dea's contract "was pretty ridiculous" and revealed that the defender had received an offer from Ukraine "that he couldn't turn down". He added that the former Celtic defender "wanted to go".
"Darren has moved on. He got a contract from Ukraine that he couldn't turn down," Nelsen told the club's official website. "Darren's the most expensive player in the whole of the league, in terms of salary cap. It frees up some space for us. He wanted to go, it was a good opportunity for him.
"Unfortunately for Darren it's not his fault. His contract was pretty ridiculous, in terms of MLS contracts."
However, O'Dea, who joined Toronto in August 2012, contradicted Nelsen's comments when he wrote on Twitter that he did not have a choice in the matter, stressing that he did not want to leave the Canadian club.
Despite his imminent departure, the club's former captain travelled with the squad to their game against Kansas City on Saturday before flying back to Toronto to finalise his transfer. The move will free an international spot on Toronto's roster, while also relieving the club's wage budget.
The Republic of Ireland international's base salary for 2013 is $350,000 (€270,000) and he was the second highest paid player on Toronto's roster, behind Netherlands international Danny Koevermans.