Juventus' Simone Zaza celebrates after scoring
Having been appointed as Roberto Martinez's successor earlier this week, Koeman was presented as Everton's new manager on Friday afternoon, with the Dutchman gearing up for a big challenge at Goodison Park.
The Dutchman will have high expectations on his shoulders as he prepares to spend Farhad Moshiri's riches - and a new striker could be on his radar.
Question marks surround the future of both Romelu Lukaku and Arouna Kone, meaning at least one striker may be needed this summer, especially given that January signing Oumar Niasse has endured a nightmare start to life in the Premier League.
Now, Sky Italia's Gianluca Di Marzio reports that Everton have joined West Ham and Koeman's former club Southampton in the race to sign Juventus hitman Simone Zaza.
The 24-year-old arrived from Sassuolo last summer, but has mostly been used as a fourth-choice striker behind the likes of Alvaro Morata, Paulo Dybala and Mario Mandzukic.
Zaza still ended the season with eight goals from 24 games, but will be looking to play more first-team football next season, sparking interest from Premier League clubs - with Koeman no doubt familiar with him given that Southampton were keen to land the Italian.
Currently in France with the Italian squad at Euro 2016 - where he set up Eder's winning goal against Sweden on Friday - Zaza may prefer to wait until after the tournament has finished before making a decision on his future, but Everton's interest now means this could be a fierce battle for Zaza's signature.
Would you like to see Everton sign Zaza?