The 28-year-old had suspended his international career indefinitely due to a wage dispute with the Togolese Football Association.
But French boss Six is determined to try and change Adebayor’s mind, and believes his involvement in the tournament is crucial to Togo’s aspirations.
"I am doing everything I can to get him to come back to us,” Six told Le Buteur. "He's a world-class player and the Togo national team captain. He is very important to the group. I will put all my strength to get him to change his mind but he has the last word."
He added, "This is about more than just him sending a warning to ensure Togolese football moves forward."
Though Six is keen for the former Manchester City striker to take part in next month's tournament, he is still optimistic about Togo’s chances without him.
"It's clear that we will need Adebayor but don't forget we qualified without him during one of the playoff matches,” he stated.
If Togo do manage to persuade Adebayor to change his mind, he would miss Spurs’ fixtures against QPR, Manchester United, Norwich, West Brom and Newcastle.