Koeman is keen to strengthen his first-team options in the New Year when the transfer window reopens and Zenit midfielder Witsel is reportedly top of his wish-list.
The Belgium international was linked with a move to Goodison Park last summer but a move failed to materialise. He has remained on Koeman's radar since and reports say a fresh offer for the 27-year-old could now be on the cards.
Spurs and Liverpool have also been linked with Witsel, as well as Juventus and Roma in Italy, but Everton look to be at the head of the queue for the £20million-rated player.
Koeman is also after more forward options and cover in defence as he plots his January transfer plans in an attempt to improve Everton's fortunes in the second half of the season following an inconsistent campaign so far.
Meanwhile, reports continue to link Paris Saint-Germain with a big offer for Romelu Lukaku in January but, while the striker's long-term future looks uncertain at Everton, a move away next month seems highly unlikely.