The Austrian, 43, orchestrated the Reds' remarkable 'Great Escape' from relegation in 2019-20, prompting interest from several suitors.
Watford were heavily linked with a move for Struber before opting for, with the 43-year-old also a target for clubs on the continent.
But Struber, despite revealing his ambition to coach at 'the highest level possible' one day, insists it is 'not the right moment to go' and is relishing his first full season at Oakwell.
Writing in the club's match-day programme, Struber, appointed by the club last November after moving from Austrian Bundesliga outfit Wolfsberger, said: "In the last few weeks, I had many offers but it's not the right moment to go.
"I have a very good team here and good mates on my side with quality - my new assistant coaches and my whole staff have good quality and high competence.
"It is now the time for a big attack on the league with Barnsley.
"My goal is to work in the highest league at the highest level possible. I think I've grown up my personal philosophy and my personal leadership in a direction and I have the feeling I can work at the highest level.
"That is my big goal."