Koné, 25, had been linked with a switch to Chelsea in the closing days of the window as the Premier League club searched for midfield reinforcements following Enzo Fernández's departure, but the move never materialised, according to Fotbollstransfers, which cited comments to Sky Sport Italia.
"Before the match against Fiorentina, I said it would be difficult for us to deprive ourselves of important players so near to the end of the transfer session, and so it proved," D'Amico said.
"Koné is an important player, we're happy he stayed with us, and we're confident he'll give his contribution during the season."
The French international joined Roma from Borussia Mönchengladbach in the summer of 2024, initially on loan before the move was made permanent, and has since become an established figure in Gian Piero Gasperini's midfield.
He scored twice and added three assists in 29 league appearances last season and has started all three of AS Roma's Serie A fixtures this term, helping the club to a perfect start that leaves them level with Inter Milan at the top of the table.
Koné signed a contract with Roma running until 2029, giving the club a strong hand in resisting outside interest. He was also part of France's squad at this summer's World Cup, where he became a regular fixture in central midfield following an injury to Aurélien Tchouaméni, and has now made 19 senior appearances for Didier Deschamps' side.
Head coach Gian Piero Gasperini was widely reported to have opposed Koné's exit, though D'Amico has since played down suggestions of a formal veto, insisting no official offer from Chelsea ever landed on the table. Roma underlined their faith in the squad on Saturday, coming from behind to beat Atalanta 2-1 at the Stadio Olimpico with two goals in the closing minutes.
Chelsea, meanwhile, ended the window without the midfield addition they had been chasing after missing out on a separate deal for Monaco's Lamine Camara, whose move collapsed at the last minute. Xabi Alonso's side, who beat Fulham and Brighton before their derby defeat at Arsenal, will be back in action against Hull City this weekend.
Koné's rise has been rapid since his breakthrough at Toulouse, where he made his professional debut in 2019 before a move to Germany with Borussia Mönchengladbach two years later.
His performances for Roma have made him a mainstay of France's midfield, and reports in his homeland suggest interest from elsewhere in Europe is unlikely to fade even with his contract running well beyond this season.