Chelsea moved for the midfielder in the final hours of the window, having identified him as an ideal replacement for the departing Enzo Fernandez, but D'Amico was clear in his press conference that the approach carried little weight.
"It was more of a probe than a negotiation," D'Amico said. "There was contact with Chelsea, but it was purely informative. We bought six players by sending initial offers, that's what a negotiation looks like.
"So Chelsea made enquiries, but never made an offer. The coach didn't give any ultimatum because there was no official offer. A week ago I had already said it would be difficult to make a major sale with so little time left before the window closed."
The comments directly addressed reports suggesting head coach Gian Piero Gasperini had issued an ultimatum over Koné's potential departure, with D'Amico making clear no such situation ever developed given the absence of a concrete Chelsea proposal. The exchange leaves Koné at Roma for the season, with Chelsea instead needing to find alternative solutions in central midfield following Fernandez's £125million move to Manchester City.
D'Amico also addressed AS Roma's unsuccessful search for a left-sided attacker throughout the summer, a position Gasperini had identified as a priority from the outset of the window.
"In the final days we had identified two or three suitable profiles, players we thought could strengthen the group," he said. "For various reasons we simply couldn't close the deals. Leweling was practically done, both with VfB Stuttgart and with the player, then he had a change of heart and we stopped. There were other players but they didn't convince us."
The sporting director also explained why a move for Luiz Henrique collapsed despite an agreement in place.
"There were issues that prevented us from completing the operation," he said. "We had an agreement with Zenit and with the player, but for technical reasons we didn't go ahead. That's why you have to be ready to find alternatives."
Asked about the decision to pursue Santiago Castro after missing out on Crysencio Summerville, D'Amico said Roma had been working multiple targets simultaneously.
"When we were on Summerville we were also on Castro, on Molina... At a certain point you have to recognise whether you can close a deal. We were strong on Summerville but also on Castro, there it became a question of timing. We then managed to close Mora, while we couldn't close Leweling, Luiz Henrique or the other names that came up, which were all real."