The Toffees have again been linked with the United States international but Arena is adamant Donovan and Keane - along with defender Omar Gonzalez - are going nowhere.
Arena told Galaxy's All-Access podcast on LAGalaxy.com: "Our players aren't going anywhere.
"Even though we argue that the [off-season] lay-off is going to be too long, he (Gonzalez) can go in camp with the US national team in January, along with Landon Donovan.
"Our players have had a really tough run this year, in particular Robbie [Keane] and Omar with all their requirements with national teams.
"They've had a full 11 months of play and they need some time off and then they need to get back here in LA and be ready for a challenging campaign that includes in the beginning not only the league but the [CONCACAF] Champions League."