Wright is filmed apparently being given an envelope of money in return for allegedly helping persuade Barnsley to sign players from a fake Far East firm.
The newspaper claims he accepted £5,000 at a meeting in Leeds in August.
The Championship club say he has been suspended “pending an internal investigation into these allegations”.
The Daily Telegraph’s investigation involved Wright being introduced to members of the Far East firm, who were undercover reporters, by two football agents.
“I can just recommend players to you that I’ve gone and seen, and you will have to do your spicy dealing, whatever you do,” Wright is filmed saying during one of a number of meetings.
Wright says “you know where I live” when the subject of giving him money is raised.
He tells a member of the firm “cheers, just put it there,” when a person hands him the envelope before the newspaper claims he left with it in his pocket.
The article makes it clear that there is no suggestion Barnsley were aware of Wright’s actions.
Meanwhile, a spokesman for Mr Wright is quoted as saying: “Any suggested acts contrary to criminal law or those of the FA and Fifa are categorically denied.”
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.
In a separate meeting, QPR manager Jimmy Floyd Hasselbaink is filmed apparently negotiating a fee to travel to Singapore to speak to the Far East firm.
The Dutchman also allegedly discusses the possibility of signing players from them.
He has not been suspended by the Championship club who say “there will be a thorough internal investigation regarding this matter”.
QPR’s statement added: “However, we have every confidence in our manager and the robust systems and processes the club has in place.”
Former Chelsea striker Hasselbaink also issued a statement in which he denied “any accusations of wrongdoing on my part”.
“I was approached by Mr McGarvey and Ms Newell of The Telegraph purporting to be players’ agents. They offered me a fee to make a speech in Singapore.
“I do not see anything unusual in being offered to be paid to make a speech.
“I did not make any promises in return. I did not ask QPR to purchase any of the players who were said to be managed by Mr McGarvey and Ms Newell and did not and would not recommend the purchase of a player for my personal gain.”
Controversial Leeds owner Massimo Cellino was also filmed by the Telegraph offering undercover reporters posing as an investment firm a way to get around FA and Fifa third-party ownership rules.
In a meeting at Leeds’ ground, arranged by football agent Pino Pagliara, Cellino apparently offered to sell shares in the Championship club as a means of funding the purchase of players.
The Italian proposed the fictitious firm buy 20% of the club, in return for which it would receive the same percentage of future player sell-on fees.
In the video, Cellino says: “I tell you, I spend eight million this year… on new players.
“You want to finance that? You want to come 20% in that? You got 20% of the player – it’s the only way.
“As a shareholder you can finance the club, asking everything you want – percentage – you are allowed to do it in England.”
Leeds claimed the footage of their owner amounted to a “non-story” as Cellino “made a perfectly proper suggestion which is entirely consistent with the FA’s regulations”.
The statement added: “If a company commits money to a club by way of investment, taking on the potential for profit but also the risk for loss, then that is a normal, everyday corporate process.
“This is plainly not a suggestion as to how to circumvent the rules, but rather, an accurate albeit concise explanation of how to operate within the confines of the rules and effectively become ‘the club’.”