Portsmouth became the first Premier League club to go into administration earlier this season and have debts of more than 100 million pounds.
They were docked nine points by the Premier League and were eventually relegated after finishing bottom of the league.
Having sold most of their leading players, they defied the odds to reach last weekend's FA Cup final where they lost 1-0 to champions Chelsea.
Grant originally joined Portsmouth as director of football in 2006 and, after a spell in charge of Chelsea, returned in the same role last October.
After manager Paul Hart was sacked in November, Grant took over the managerial duties and was in charge throughout the chaos of administration.
The club announced his resignation on their website on Thursday and Grant then posted a letter to the fans.
"This was one of the most difficult decisions I've had to take in my football career," said Grant, who has been widely tipped to take over West Ham United after they sacked Gianfranco Zola last week.
"After a lot of soul searching and under the circumstances I'm taking a different direction.
"It's been both a difficult and complex year for us at the club, but at the same time, it's been a wonderful and uplifting professional and personal experience.
"I have been inundated with letters and emails from fans. Many have brought tears to my eyes and take it from me; it takes a lot to do that.
"I will never forget you, the loyal fans of Pompey who, without a doubt, helped me protect the team under such complex circumstances. There are very few teams in the world that have fans as passionate and devoted as you are.
"It has been a great experience to be part of one of the most wonderful occurrences that has happened in the history of British football and also shows that even in the most desperate times, loyalty, devotion, professionalism and passion can be demonstrated.
"Portsmouth has given me a feeling of home away from home. I might be leaving Portsmouth physically, but you cannot take Portsmouth away from me and my heart," Grant said.
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