Kondogbia, who has been linked with a move to Arsenal and Tottenham this summer, is set to sign a five-year contract with a transfer fee of around ??22million, according to Sky Italia.
The Frenchman looked set to be joining AC Milan until Inter negotiators arrived in Monte Carlo on Friday to speak with the 22-year-old.
Kondogbia was one of Monaco’s stand-out players as the Ligue 1 side reached the Champions League quarter-final, in particular impressing in the 3-1 win at Arsenal in February.
Inter struggled in Serie A last term under Roberto Mancini, finishing eighth in the league and being knocked out of the Europa League at the round of 16 stage.