Rooney trained with his new team mates at their Finch Farm centre earlier in the day before addressing the cameras with Koeman when he said returning to the club was a "no-brainer".
Koeman immediately made clear that he expected Rooney to provide an offensive threat, ending suggestions that he might play in midfield.
"Everyone knows the qualities of Wayne. We're trying to get more productivity. He can be 10 behind the striker, the 9, the 7," said Koeman.
"We need to improve our goal strike rate. Last season Romelu (Lukaku) scored 25 goals and the next best was with four or five. That's not good enough. Wayne is the sort of player who can bring productivity to the team.
"Wayne was desperate to come back. He showed us in the first meeting that he likes the pressure. Some players need that to get the best out of themselves."
Koeman confirmed Lukaku is about to sign for United and presented Rooney with his favoured number 10 shirt before the news conference.
Rooney said he was delighted to be back at his boyhood club and intended to help them win their first major trophy since their 1995 FA Cup triumph over Manchester United at Wembley, a game he attended as a supporter.
"All my family are delighted to see me playing in a blue shirt again," said Rooney, adding that he had not contemplated a return until a lack of first-team opportunities at United forced his hand.
"Not being in the team at United was frustrating. My whole career I have played, that was then first time I was on the bench. It's not me. I need to play. But obviously I was captain there so I had to be positive.
"I never really thought about (returning to Everton) that much, only when I was at the stage when I had to take the next step.
"Once I knew Ronald Koeman was interested in bringing me back to Everton it was a no-brainer, it's where I wanted to play."
Rooney, who will be 32 in October, has signed a two-year deal at the club who finished seventh in the Premier League last season, one place behind United.
"I'm excited, it's a challenge which is the right time in my career. Hopefully, we will win trophies. This isn’t a retirement home," he said.
(Reporting by Neil Robinson, editing by Ed Osmond and Christian Radnedge)
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