Moyes is looking for additional cover at the back and sees the 25-year-old international - who was the subject of interest from Tottenham in the summer - as a good fit for his side.
Marseille have offered Taiwo a new contract but his current deal expires in the summer so he is now free to negotiate a move away with interested clubs.
Let battle commence: A number of clubs are in the hunt for Taiwo
The best offer so far has come from Benfica, who want him as a replacement for Fabio Coentrao.
Coentrao is primed to leave will go in the summer and Manchester United, Chelsea, Bayern Munich and Real Madrid all want him.
The Nigerian left back has been with Marseille since 2004 and his 39 appearances for the Super Eagles automatically qualifies him for a UK work permit.
Moyes knows that funds are, as ever, limited this January and the club may struggle to meet his wage demands.
The former Lobi Stars man is believed to be very interested in a move to England but Fenerbahce are also rumoured to be weighing up a bid. The Friday Five: Memorable clashes between Liverpool and EvertonLet battle commence: Red-hot derbies and White Hart glamourRooney and Henry pairing at New York Red Bulls after MLS draft pickAll the latest Everton news, features and opinion
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