Spalletti sealed Inter's Champions League berth for a second consecutive year with a 2-1 win over Empoli on the final day of the Serie A season Sunday.
Inter snatched fourth place, equal on points with Atalanta, but behind the side from Bergamo on head-to-head difference, to edge city rivals AC Milan by just one point in the race for elite European football.
But multiple reports state that the 60-year-old former Roma and Zenit St Petersburg coach will be axed in favour of ex-Juventus, Chelsea and 'Azzurri' boss Conte.
Gazzetta dello Sport said "total agreement" had been reached with 49-year-old Conte, with Corriere dello Sport and Sky Sport Italia reporting a three-year deal would be sealed before the end of the week.
Spalletti conceded his future was uncertain, despite returning the club to the Champions League after a seven-year absence this season, where they were knocked out at the group stage.
"This result (against Empoli) changes little. The club will talk now and I will listen, just as you will," Spalletti said after Sunday's game.
"I would like to continue to coach Inter. The newspapers have had the same headlines for three months, so we'll see what happens."
Conte has also been linked with a return to champions Juventus, after his successor in Turin Massimiliano Allegri was removed, and AS Roma where Claudio Ranieri has been the interim coach since the sacking of Eusebio Di Francesco in March.
Conte won the first three of Juventus' eight straight Serie A titles and the Premier League and FA Cup with Chelsea.