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.
On the situation, Heckingbottom said: “I cannot talk about it. It is just a football press conference and the club and myself will make statements later as there are still things ongoing.
“We have to do things properly and speak about things in the right time and right way as it is not nice for anyone.
“Everyone has seen that Tommy has been suspended due to the ongoing investigations and allegations and that is as far as we have got.
“Things are always rolling. But my focus is on the players and the team and that it is.
“There will be a time for me to speak about it later on. At this moment in time, everything I am doing is focused on the players and team, just as it always is.
“I will tell the players that I can’t let my focus slip and that’s my job.”
Heckingbottom says that developments with Wright will affect plans for the weekend game at Elland Road, with a meeting to be held this morning with players about the situation following the suspension of the Reds’ No 2.
He said: “The plan is different. I had one thing sorted this week and it goes out on the window. My focus is the players and on them being alright and comfortable with what is going off and understand their role going forward. Then it will be a case of training and me watching them and chats and conversations.
“My role this morning in terms as a coach will not be tactical things, but more personable things and making sure everyone is okay.”