Arnold played the lead in the 78-year-old filmmaker's 1990 science-fiction classic Total Recall, and Paul wants to team with him again, according to editors at British newspaper The Times.
In particular, the Elle director has his eye on helming a follow-up to the 1982 and 1984 films Conan the Barbarian and Conan the Destroyer, directed by John Milius and Richard Fleischer respectively, in which the action star played the titular character.
Criticising the films ex-governor of California Arnold has made since he returned to acting full-time in 2011, Paul explained, "The scripts he's done recently are not representing the best of Arnold. The fans are always saying they want me to make a Conan the Barbarian sequel."
Since an unsuccessful 2011 reboot unrelated to the original films, a new Conan film starring Arnold has been in the offing, with a draft script having been written by scribes Will Beall, Andrea Berloff and Chris Morgan.
While promoting another action sequel, Terminator Genisys, in 2015 Arnold revealed the basic outline of the script for the new Conan movie.
"I was hoping to do another one, and I (liked) the idea of Conan having been king for a long time, then just threw it all away and went into retirement and went off into the mountains," he told reporters at a press conference.
"That whole idea always appealed to me. Then of course he gets asked back because of some hideous and unbelievable things that are happening to the kingdom. So that's when the battle begins again."
There is currently no director attached to the new Conan film, which has not received an official release date.
Paul, whose last film Elle was named in many critics best of 2016 lists and triumphed at France's Cesar Awards, is also interested in making a comedy with fellow veteran filmmaker Woody Allen.
"If Woody Allen would give me a script I would do it," he added.
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