Inzaghi won a Serie A title, two Coppa Italias and three Italian Super Cups during his four years at Inter but could not deliver the Champions League title and left the club just days after their humiliating defeat by PSG in Munich.
"It might look like it's something that came quickly, but it's the result of hard work," Al Hilal CEO Esteve Calzada said.
"He was playing a massive match and asked (us) to keep things aside until after the final, it was decided but not signed before the final, just because out of respect he asked us to wait, which is certainly fair enough."
The comments prompted Italian media to question whether the Inter squad knew about the move ahead of the final and if it had any impact on their performance.
Inzaghi, speaking ahead of Al Hilal's first game at the Club World Cup against Real Madrid on Wednesday, said there could be no questions about his commitment to Inter during his time there.
“Today I heard - or heard about - all sorts of things, as has often happened during the four years I was at Inter,” the Italian told a press conference.
“If this was the price to pay for my four years at Inter, I’ll gladly pay it. But it’s nothing compared to the good I received from the entire Inter world - I mean the fans, the management, the players, everyone at Inter.
"I know I’ll miss it - I’ll miss everything, even this - even the most unfair accusations that were made over these four years. But I was truly happy. I gave my all."
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Inter President Beppe Barotta did not want to be drawn into the controversy but said no individual was bigger than his club.
"I don’t chase after rumours, denials, or other people’s behaviour - everyone knows what they’ve done,” he said before Inter's Club World Cup game against Monterrey on Tuesday.
“A cycle has ended and we thank Simone Inzaghi for all he’s given us, but now there’s a new cycle to open and focus on. Inter’s story moves forward and doesn’t depend on individuals. "