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Skriniar was strongly linked with a move to Paris Saint-Germain last year, but no move materialised and the centre-back began renegotiating a new deal with Inter.
However, his agent confirmed on Tuesday that Inter's offer was ultimately rejected in December, and he has since made clear to the club they are in talks with other teams.
Skriniar, who was sent off in Tuesday's (AEDT) 1-0 defeat by Empoli, is widely expected to eventually sign for PSG, though Sistici refused to name the clubs he is speaking to.
"The truth is that the decision to place Milan Skriniar on the transfer market in the summer was made by Inter, obviously not by the player," Sistici said.
"It was a choice by the club that led to a negotiation between Inter and PSG, of which we had obviously been informed. At a certain point, the negotiation broke down, and we had also been informed of this step, a decision that was not up to us and which the player accepted calmly, with professionalism and absolute respect for the contract.
"Last autumn, we responded to all the meeting requests made by the club. After a series of preliminary meetings, we presented our financial request and subsequently, at the beginning of November 2022, the club presented us with a proposal.
"About a month later, before Christmas, I communicated to Inter the decision not to accept their offer, a choice reiterated at the beginning of January, before the Supercoppa, when I also informed the managers [Giuseppe] Marotta and [Piero] Ausilio that we thought we were free to listen to offers from other clubs.
"Maybe it wasn't mandatory communication, but for the player and myself it was the most correct way to proceed, knowing that seriousness and transparency are the best way to proceed for us. I repeat, we have never missed any appointment with Inter."
He said: "We didn't talk about PSG, just as we didn't leak our and Inter's positions, given that with Inter we had agreed to handle the matter with the right confidentiality, in the interest of serenity and the performance of the team and of the player.
"There are contacts with some clubs. Not Italian [clubs]."
Inter coach Simone Inzaghi responded after the shock loss by saying the club were "trying to resolve" the situation with Skriniar, without clarifying whether he meant by selling the player or making a new contract offer.
At a similar time, PSG coach Christophe Galtier was also asked about Skriniar following the Parisian's 7-0 Coupe de France win over Pays de Cassel.
He did not offer much clarity either, though he acknowledged PSG was convinced last year they had done enough to sign him.
"I cannot say if he will arrive this winter [January], or this summer," he said "He is a player we've been tracking for a long time.
"In fact, we thought that we had already got him last summer."