Uruguay international Godin announced his decision to bring his nine-year stay at the club to an end in an emotional media conference last week, when he revealed he and the club "have not reached an agreement to continue".
Simeone, who was visibly upset while listening to Godin reflect on his Atleti career, indicated that he is unhappy with the failure to keep the 33-year-old.
"The club and I wanted Diego to stay," Simeone said ahead of a home meeting with Sevilla.
"But Diego, looking at his circumstances and the relationship he has with Atletico, took a decision that he explained few days ago."
He added: "There's nothing more I can say about Godin.
"I always valued his presence in this club, but as I said the management and the players have different things to achieve and Diego already expressed his feelings."
Godin's departure and anticipated move to join Inter Milan in Serie A prompted criticism from former Atletico captain Gabi, who said the club's contract policy for players over the age of 30 was "unfair".
Simeone responded to Gabi's comments, saying: "Well, the club has a way to do things and we must respect it.
"Those of us inside the club have to agree with the decisions of the club, and we get behind them.
"As with every situation in life, some people make the rules and others have to follow them."