The Scot led the Shrimps to a historic first-ever promotion to the third tier in Monday's League Two play-off final with Newport County at Wembley.
It came amid the backdrop of long-standing interest from City in the Aberdonian, 45, who first spoke with the club about the position in December and was interviewed.
Adams's promotion record, specially from football's basement which has seen him go up with both Morecambe and Plymouth, saw him swiftly emerge as City's number one choice.
Adams had another year left on his contract at Morecambe and City would have to agree compensation to complete his move to Yorkshire.
A statement on the Morecambe website read: "Derek Adams has today left Morecambe Football Club to pursue an opportunity elsewhere."
Co-chairman Rod Taylor commented, "Derek did a fantastic job during his year and a half at Morecambe, not only guiding the club into League One, but also acting as a catalyst for positive change and encouraging everybody at the club to aim higher.
"For all of that we thank him, and we wish him all the best for the future.
"However, the club is now solely focused on moving decisively to appoint the right manager to build upon the huge opportunity that promotion creates for both the football club and the town.
"That manager will be tasked with delivering the club's ambitious vision and strategy, and conversations have started already."
Meanwhile, City have announced that a press conference to unveil the identity of their next manager will take place on Friday afternoon.