The 55-year-old was relieved of his dutie s on Thursday, just 14 months into his second spell at Oakwell.
Barnsley cited the club’s League One position as “unacceptable” as one of the principle reasons why Wilson was dismissed.
Former favourite Craig Hignett – released from the coaching staff at Middlesbrough only last month – has been linked to the vacant post but the club are keen to keep their options open and their cards close to their chest.
“We are keeping our own counsel and not putting a time frame on it,” said Reds chief exectuive Ben Mansford.
“Whoever we bring in has to have the aspirations of getting us back into the Championship.”
Wilson celebrated his 1,000th game as a manager before Christmas but lasted just nine further fixtures before receiving his marching orders.
Despite this spell being deemed a failure, his first at Oakwell will be fondly remembered by all at the club as he steered them into the Premier League in 1996-97.
Mansford said: “It was a very difficult decision to make given the professionalism Danny has conducted himself with, not only in his second stint with Barnsley, but throughout his entire career.
“Personally, I couldn’t pay Danny a big enough compliment.
“He came in and felt like he could achieve survival last year, and in the end it just wasn’t to be.
“Danny will always be welcome back at the football club and he goes with our thanks.”
Barnsley are at Crawley on Saturday, a team who hinted at the challenges Wilson and a team awash with new faces would face in League One when they won 1-0 at Oakwell on the opening day of the season.