The Temple of Satan argues the First Amendment to the U.S. Constitution means if one religious monument goes up at a government building it should be open to everyone, CNN reported.
"They said they wanted to be open to different monuments, and this seems like a perfect place to put that to the test," said Lucien Greaves, a leader of the group.
The Ten Commandments monument was approved by the state Legislature in 2009 and erected with money donated by one lawmaker.
Greaves said he has the application for the monument and is working on a design, hoping for "something big and bold" and strong enough to survive attacks.
"My favorite idea right now is an object of play for children," he said. "We want kids to see that Satanism is where the fun is."
The temple's website lists nine principles involving respect for others' freedom, showing compassion and other virtues.
The group has engaged in actions such as a same-sex kiss-in at the grave of the mother of the Rev. Fred Phelps of the Westboro Baptist Church, which has become notorious for its anti-gay demonstrations. It also rallied in support of a Florida law allowing "inspirational messages" at school assemblies -- the temple said it was planning to take advantage of the law.
State Rep. Mike Reitz, the Republican who paid for the Ten Commandments monument, declined to comment, telling CNN he is involved in litigation with the American Civil Liberties Union. A fellow Republican, state Rep. Bob Cleveland, said the Satanist monument should not be approved.
"I believe that only monuments that reflect Oklahoma values should be allowed on Capitol the Capitol grounds," Cleveland said in an email Monday.