The 31-year-old has not played for Tottenham this season and had been training with the reserves, having been left out of both their Premier League and Europa League squads.
Adebayor had a year left on his contract and had been seeking a £5m pay-off to walk away prematurely.
“On the professional side, we needed to split,” said Pochettino.
“It was clear from last season, four months ago, the situation was clear.
“I’m happy because for him, it is important that now he has sorted the problem and as a club, we wish all the best to him because our relationship for one year was very good.”
Aston Villa’s former Tottenham boss Tim Sherwood, who temporarily revived Adebayor’s career while in charge at White Hart Lane, has been linked with a move for the forward in the January transfer window.