Dzeko, 35, scored in a 3-0 friendly win over Dynamo Kiev on Saturday and is seen as a replacement for Romelu Lukaku at Inter, following the Belgian star's return to Chelsea, and leaves Roma after a successful six years at the club.
The Bosnian striker joined from Manchester City in 2015 -- initially on loan before the move became permanent 12 months later -- and netted 119 time in 260 appearances. He was the league's top goal scorer 2016-17 with 29 goals.
Dzeko's future has been uncertain since a falling out with then-Roma manager Paulo Fonseca, who was replaced by Jose Mourinho.
- Don't have ESPN? Get instant access
Dumfries, meanwhile, impressed at Euro 2020 for Netherlands, scoring two goals. The 25-year-old leaves PSV after three seasons at the club.