The club's two highest earners have no place in Antonio Conte's side, CEO Giuseppe Marotta told Sky Sport Italia on Saturday.
"In a project, we must find the right profiles. This is the path we are taking," Marotta said in Italian. "We have been able to speak clearly with those directly concerned while respecting their contractual and professional values. But we have explained the position of the club. This is the most important element.
"Both of them are aware that this situation does not mean their abilities have diminished. They are both great players and great talents. But while talent alone wins games, it is the team that wins the championship. As a result, they are not part of our project. I say this with transparency and respect."
Icardi, 26, was ridiculed by sections of supporters and stripped of the captain's armband during a tumultuous 2018-19 season in which his wife and agent, Wanda Nara, openly feuded with Marotta and sporting director Piero Ausilio over a new contract.
Conte, who was appointed manager of Inter in May, then decided to exclude Icardi from his plans. The Nerazzurri are reportedly targeting Manchester United striker Romelu Lukaku as a replacement for the Argentine.
Nainggolan is on the market after just one year with Inter. The side acquired the divisive Belgian international last summer in a deal that sent Nicolo Zaniolo to Roma. Inter suspended the 31-year-old midway through the season for disciplinary reasons.