The Foxes have lost all of their eight league games so far this season and sit bottom of the Women's Super League.
Morgan has played a big part in the club's history, helping to guide Leicester from Midlands Division 1 to the WSL.
The Foxes' assistant manager Michael Makoni and first-team coach Holly Morgan, Jonathan's sister, have also departed the club.
A club statement said: "Retention of its FA Women's Super League status is central to the club's vision for the long-term development of women's football in the city.
"The club feels a change in leadership is now necessary to give the team the best possible chance of achieving this in the remaining months of the season."
Jonathan Morgan added: "This club has been part of mine and my family's life for 12 years from when my sister Holly Morgan joined as a young player.
"To have taken Leicester from the Midlands Division 1 to an FAWSL club gives me nothing but pride.
"Earning promotion last season in the manner we did will always be a memory I can share with the fans and the city.
"This season has been disappointing, with injuries contributing to a difficult start, but we haven't achieved the results we had hoped for. I wish the club, the staff and the players my very best."
Heskey, 43, will take charge of the club's training sessions ahead of their Continental Cup clash against Manchester United on December 5, while the club begin their quest to find Morgan's successor.
He has held the position of head of women's football development at Leicester since October.