O'Donoghue mentored Bruce to the final of the BBC contest and was later said to have dumped his girlfriend Irma Mali in order to begin a romance.
Though O'Donoghue has denied that he is dating Bruce, the rest of the 'Breakeven' band are allegedly concerned that the furore could overshadow their work.
"They feel all the speculation could have a negative impact on the band," a source told Now. "He needs to focus on what's important now."
The insider also claimed that O'Donoghue feels he could have done more to prevent the stories about him and Bruce from causing extra hurt to Mali.
"Danny's upset that he did little to help an already fragile situation with Irma," they said. 'He was just trying to be a good friend and mentor to Bo."
O'Donoghue also took to Twitter shortly after Leanne Mitchell, mentored by Sir Tom Jones, won The Voice earlier this month to blast the press for diminishing her victory with their "silly stories".
"Focus should be on Leanne and the brilliant job she's done," he wrote. "Well done team Tom, what a Legend;)."
The Script are currently working on their third studio album, which will follow 2010's Science & Faith and their 2008 self-titled debut.