After defeat to Crystal Palace, the Terriers became only the second club to go down this early in a season, matching Derby's dismal record in 2007-08.
When asked whether his intention was stay on, Siewert answered: "For me it's important to go through all the details — to make the club stronger is not just about going down directly or going up.
"You never know these things in football. You can't promise it. But the thing is I'm willing to work hard so that we come back to the Premier League.
"Because I know the club that joined the Premier League for two years really enjoyed it, and our supporters really enjoyed it.
"And for me, of course, as a manager if you are ambitious, of course (I would stay)."