The former Leeds United winger was linked with a reunion with Lee Johnson at Bristol City, with the pair having previously worked together at Barnsley and Oakwell.
But head coach Heckingbottom says that Wright, 50, will be remaining at the club.
Heckingbottom, who says that he is hoping to sign four more players before the start of the new season, said: “Tommy is staying. For me personally, it’s a great signing as he has been fantastic for me and the players and the staff love him.
“It was literally at the final whistle (at Wembley) that I was speaking to Patrick (Cryne) about having to keep the team and staff together. That is your building block and base.”
Barnsley, who have completed the signing of George Moncur from Colchester United on a three-year deal, are expected to seal the capture of West Ham forward Elliott Lee tomorrow.
On Moncur, 23 in August, who has signed for an undisclosed six-figure fee, Heckingbottom said: “George had a number of suitors, so it is a pat on the back for us as a club that he chose to come to us.
“We know his abilities and know he can play as a striker or central midfielder and that’s where we predominantly see him.”