McLeish was appointed manager at the City Ground on December 27, but cited "a difference in the understanding of the development strategy of the football club."
He was reportedly unhappy at the collapse of George Boyd's transfer from Peterborough on deadline day.
Peterborough chairman Darragh MacAnthony said Forest pulled the plug on the deal, claiming Boyd failed his medical due to poor eyesight.
McLeish said in a statement released by the LMA: "I am extremely saddened to be leaving after such a short period but even so it has been a privilege to be manager of such a fantastic football club and I want to thank the players, staff and fans for their support and wish Nottingham Forest the best of fortunes ahead.
"Both parties entered into the relationship in good faith, however, there was a difference in the understanding of the development strategy of the football club and it was felt by both parties that we should part company."
Forest chairman Fawaz Al-Hasawi said: "I’m bitterly disappointed – as is Alex – that it has not worked out as we had hoped when he became our manager and I would like to put on record my thanks for his services to the Club and wish him every success in the future."