The 35-year-old Brazil international's representative, Paulo Tonietto, says PSG sporting director Leonardo initially did not want to keep him in the French capital, but ended up turning back on that decision by proposing a new contract after the agreement with the Blues was reached.
"He had given up hope [on a new contract offer] because Leonardo told him two months ago that he did not want to extend it," Tonietto told L'Equipe. "But last Tuesday, Leonardo called Thiago and raised the possibility that he would stay another year.
"Thiago had already reached an agreement with Chelsea, there was no turning back. When you make such a proposal, you have to be convinced. A sporting director who tells a player he doesn't want him anymore, and two months later wants to extend it, how do you take that?"
Tonietto says that PSG coach Thomas Tuchel had hoped to hold onto the former AC Milan star, but there was no chance of him agreeing to stay after Leonardo said his time in the French capital was up.
"In a club, the sporting director and the coach have to walk together. This was not the case," he added. "Tuchel always wanted to keep Thiago. Leonardo was the one who didn't want to, and he's the one with the power.
"That's why, even if he hadn't signed with Chelsea, Thiago couldn't see how he could have accepted Leonardo's proposal given what he had told him two months earlier. It would have been humiliating.
"We never talked about money or the length of the contract. Never. Leonardo just said he wasn't counting on him anymore. Nothing more."
The Blues confirmed the arrival of Silva on Friday, with the defender signing a one-year deal with an option for a second season.