"I am happy to be going to this great team," he said at a farewell ceremony after Sevilla confirmed it had sealed his transfer to the English Premier League runners-up.
"I have had four very happy years with Sevilla, but there are stages in life and mine here has finished."
The club confirmed his transfer, subject to a medical, in a statement earlier.
It declined to say how much City paid for Negredo but Spanish and British media reports put the sum at around €28 million ($37 million).
Sevilla's president Jose Maria del Nido said the sale would bring "a stack of millions of euros to Sevilla's coffers, which will enable us to grow".
Negredo said he was convinced to make the move by a call from City's Chilean manager Manuel Pellegrini, who joined the Manchester club from Malaga last month.
"He totally convinced me. He told me what they could offer me there and for me it was a new challenge."
Spain international Negredo, 27, follows winger Jesus Navas, signed last month from Sevilla by the English club.
"Manchester City is buying a magnificent player, the top Spanish goalscorer in the Spanish league," Del Nido said, referring to Negredo's 25 Liga goals last season.
"Sevilla is getting a great deal and the player is signing the contract of his life, so I think we should all be happy."