Ginnelly claims the incident happened as he was substituted in the latter stages of their 3-2 defeat.
"It's not acceptable in the game - and it's not acceptable to be spoken to in that way by anyone," said Tolson.
The Football Association are conducting an investigation after match officials reported the 18-year-old's complaint.
"The biggest concern for me as a manager - how is Josh and how is it affecting him?" Tolson told BBC Sport.
"It's unacceptable to anyone, not just our player. That is across the board."
Ginnelly, who joined the club on loan from Burnley in January, reported the incident to fourth official Natalie Aspinall and later made a statement in the presence of representatives from Altrincham and the police.
Barrow chief executive Austin Straker said the club was also carrying out a review of available video footage.
"The club have no recent history of such behaviour and are appalled by the possibility that it may have occurred," he added.
"There can be no place in any community for any form of discrimination, and every effort is made at the ground and in the local affiliations to educate fans to this end.
"The club have an excellent record of working with the authorities in dealing swiftly and firmly with any unacceptable behaviour in, or indeed around, the stadium."
Altrincham caretaker boss Neil Tolson says alleged racist comments directed at midfielder Josh Ginnelly in their defeat at Barrow were "unacceptable".