The pair have both had loan spells in England in recent years with Keane spending two months at Aston Villa at the start of 2012 while Donovan joined Everton on loan in 2010 and 2012.
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 [offseason] layoff 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."