“I’ve had some really positive conversations with the board and James Ellis, who have been very clear about the club’s ambition.
“I’m ready for the challenge ahead and looking forward to meeting the players, staff and supporters.
“Preparations for pre-season are well under way and we have an important and exciting few months ahead.”
Skubala won the League Managers’ Association League One manager of the season after taking Lincoln to the second tier for the first time in 65 years by winning the title.
“Michael has made a significant impact during his time at Lincoln City and leaves with our sincere thanks and best wishes,” Lincoln chief executive Liam Scully said.
“From the moment he arrived, he embraced everything about this football club and represented the club with real integrity and professionalism.
“We are proud that Lincoln City continues to be recognised as a stable place where talented people develop and Michael leaves with the gratitude and respect of everyone connected with the club. I’d like to thank him for everything he has contributed and wish him every success in the future.”
His appointment is expected to cost Bristol City more than £1m in compensation.
Skubala is understood to have signed a new contract at Sincil Bank after the Robins initially tried to bring Bournemouth assistant Tommy Elphick to Ashton Gate.
The former Brighton and Aston Villa defender had positive talks and was impressed with the club and its personnel but decided against making the move on Wednesday.
With Marco Rose replacing Andoni Iraola at the club, sources have told BBC Sport Elphick wants to assess his options further.
Lincoln have appointed Chris Cohen and Tom Shaw as joint head coaches to replace Skubala.
