The 53-year-old manager will replace Rui Borges, with the Portuguese club announcing on Sunday that they had initiated discussions to terminate his contract.
Sporting will pay a €5 million fee, Lyon added in a statement.
Lyon said they "deeply regret" this "unexpected decision", particularly given "its timing in the middle of the season".
Sporting sit sixth in the Champions League standings with nine points and third in Liga Portugal, while Lyon sit 12th in the Europa League and third in Ligue 1.
Jorge Maciel, who was Fonseca's assistant, will take charge of Lyon on an interim basis.