Saul Anuzis, a former head of his state party, is the first serious announced rival to Steele, The Hill newspaper reported.
Steele has not stated whether he will seek a new term, but he said last week, "My hope is that folks still believe I still have something to contribute," The Washington Post reported.
Anuzis ran for the chairmanship two years ago in a divided field where Steele won.
Steele's chairmanship has been marked by occasional gaffes and lavish spending, but he is claiming credit for Republican victories in the midterm elections.
"Chairman Steele's record speaks for itself," Anuzis wrote in a memo to RNC members. "He has his way of doing things. I have mine. … But if you agree with me that victory in 2012 requires a new set of priorities and new leadership I hope you will consider supporting my candidacy."
The decision will be made in January by the 168 RNC members. Steele's aides say he has as many as 60 of the 85 votes needed to win.
Notable deaths of 2014 [PHOTOS]