Producers Wall to Wall and their owners Warner Bros Television Group will pay for the extra costs, Broadcast states.
The BBC reportedly invested £22 million in the music series for a two-year deal with John de Mol's Talpa, who own the format.
Costs for the set are apparently behind the overspend on the show. Broadcast claims that each of the show's 12 episodes cost on average £650,000.
A Wall to Wall rep said: "The Voice UK was the most successful new entertainment series launch on BBC One in more than a decade. As with any first series, it is important to get as much investment on screen as possible. The viewers obviously liked it, as the show averaged a 38.5% share."
BBC One controller Danny Cohen insisted this week that he was satisfied with the ratings and output from the first series, claiming that The Voice was a Saturday night show that had "a big impact" and which "connects with BBC values".
Photo gallery - The Voice UK Final:
Copyright: BBC/Wall to Wall/Guy Levy
Cohen did add that there were "things you learn" from a first series and appeared to hint at changes for the 2013 series.
The Voice UK was the BBC's biggest entertainment launch in over a decade with higher first series average ratings than The X Factor and Strictly.
However, the ratings did slide during the live shows to a low of 4.49 million in the semi-finals and winner Leanne Mitchell has so far struggled in the charts with her debut single 'Run To You'.
Watch the music video for Leanne Mitchell's 'Run To You' below: