Sarri, who has been strongly linked with a move to Chelsea, guided the club to a second-place finish in Serie A last term and helped them mount a serious title challenge.
Despite this feat, De Laurentiis felt that Sarri had to be replaced because he was unwilling to wait for the manager to make his mind up about where his future lies.
"There is no background to Sarri's farewell. At a certain point I told the press that the time was over," he told Sky Sports News.
"Since I am responsible for such a particular company like ours, a football company like Napoli, I can't be always waiting for the others to be available when they should always be available for the club.
"So I gave him time, but then the time was over and I moved on by taking Carlo Ancelotti to Napoli. With Ancelotti we started to collaborate both together for two days in Rome and then with him first in London, now in Canada."
Chelsea are yet to confirm the departure of current manager Antonio Conte, but reports suggest that the Blues have already made an approach for Sarri.