The report included footage of Wright apparently accepting money in return for allegedly helping persuade the Championship club to sign players.
He was hanging on Wednesday while the club investigated the allegations.
But the 50-year-old has now been dismissed with immediate effect after a meeting with chief executive Linton Brown.
A Barnsley statement read: "After considering Mr Wright's response to allegations in today's Daily Telegraph about breaching FA rules over player transfers, Mr Wright was dismissed.
"The club was unaware of such matters or involved in any wrongdoing. The club will continue to fully investigate the issues at hand and will co-operate with the regulatory authorities as necessary."
As part of an ongoing investigation into alleged corruption in football, the newspaper claims Wright accepted £5,000 at a meeting in Leeds in August.
A spokesman for Wright has said: "Any suggested acts contrary to criminal law or those of the Football Association and FIFA are categorically denied."
Wright joined Barnsley in February 2015 as number two to former boss Lee Johnson, and
continued in the role under current head coach Paul Heckingbottom.
The latest allegations come a day after the Telegraph claimed eight current or former Premier League managers had taken bribes for player transfers.
Sam Allardyce left his post as England manager on Tuesday after claims in the newspaper that he offered advice on how to "get around" rules on player transfers.
The FA, Premier League and EFL have pledged to investigate this week's newspaper allegations.
In a joint statement, they said: "English football takes the governance of the game extremely seriously with integrity being of paramount importance.
"Any substantive allegations will be investigated with the full force of the rules at our disposal, which are wide-ranging and well-developed. In addition, should we find any evidence of criminality we would inform and seek the support of the appropriate statutory authorities."
The League Managers' Association (LMA) has accused the newspaper of holding up any investigation because it has yet to hand over any evidence. It said in a statement it wanted "full and complete unconditional disclosure of all the information it has, immediately".
The Telegraph has said it will pass on all relevant material to the football authorities.
