Doak has been heavily linked with a move to Liverpool in a deal which would net Celtic just a training development fee for the 16-year-old winger.
And Postecoglou says success in the game can be achieved by staying at Celtic – with a nod to the careers of Kieran Tierney, Callum McGregor and James Forrest as examples.
"When you look at the starting line-up [against Dundee United], there's a few boys who started out as ball boys for our club, you can't tell me they haven't had careers - they've represented their country, they've won trophies,” he said, in reference to McGregor and Forrest.
"There's a couple down south who started here too. People may think going down south is a reward or some guarantee of success, life - and football in general - shows you there are many avenues towards that goal.
"I think you're in a great place here if that's what your end game is."
Doak has played twice for the Parkhead first team and has shown up well in Lowland League appearances with the B team. The lure of working with Jurgen Klopp is said to have been a temptation, according to The Athletic.
But Postecoglou, who handed the teenager his senior debut in January, told BBC Sportsound his focus was “to look after the footballers that want to be here.”
"The footballers that don't want to be here? There's another avenue there for them,” he added.