Toshack, the son of former Swansea and Liverpool legend John Toshack, is currently in charge of Swansea's Under-23s.
However, the report claims the Welshman is keen to manage at senior level and has met with Barnsley about succeeding Paul Heckingbottom.
Toshack led Swansea's U18 team to consecutive FAW Youth titles, before guiding their U23s to the Premier League 2 Division 2 crown in 2017.
Current Barnsley loanee Oli McBurnie was a prominent member of the latter side.
According to the Sun, Barnsley want a young boss and have also spoken to St Mirren's Jack Ross.
Meanwhile, the Daily Mirror reported on Wednesday that Tykes caretaker Paul Harsley has emerged as a front runner for the position, alongside Garry Monk and Gareth Ainsworth - although an approach for the latter has since been turned down by Wycombe Wanderers.
Toshack is the oldest of those, at 47.