O'Driscoll, 55, surprisingly sacked by Forest just hours after their Boxing Day win over Leeds, was offered the vacant Barnsley job following talks with the club last week, but announced in a statement on Sunday that he would not be replacing Keith Hill.
"Our intentions now are to hold further interviews obviously now that Sean has made his feelings known," Barnsley director Don Rowing told Press Association Sport.
"We're disappointed Sean did not want the job at this time, but we're respectful of his decision and his reasons why.
"We had a total of five names on our short-list, so we shall hopefully start interviews over the next 48 hours.
"It's always going to be difficult to get the man in that everybody wants. (Doncaster chairman) John Ryan has just said that the right choice for some is not going to be right for everyone.
"It's all about opinions. Sometimes clubs make a decision over a manager and it works and sometimes it doesn't, but we shall be doing our best to get the best man available to take Barnsley Football Club forward."
Barnsley sacked former Rochdale boss Hill following the home defeat to Blackburn on December 29 after a run of one win in 14 matches and assistant David Flitcroft has taken charge of the first team for the last two matches, including Saturday's FA Cup third-round win over Burnley.