The Italian scored 23 goals in 28 games last season under Koeman's management and the Dutchman is keen to link up with him again in the Premier League.
The Saints have sold Luke Shaw, Adam Lallana and Rickie Lambert for a combined £65m this summer and are showing intent to reinvest that money, with Pelle's reported £9m arrival imminent and poised to add to the acquisition of Dusan Tadic.
He told reporters: "Pelle is close but it's still not signed. We're getting closer together and we hope maybe this weekend to have him as the second new signing.
"Tadic is a player who scores and makes assists. I know him very well from Holland an it's a player who has very similar qualities to Lallana. He's a great player for us."
Pelle has previously played for Serie A sides Lecce and Parma, as well as AZ Alkmaar in the Netherlands.