Cruz has also hit South Carolina, another early voting state, attending a dinner there in early May.

"Senator Ted CruzRafael (Ted) Edward CruzWhatever you think the Alabama special election means, you’re probably wrong This week: Congress gets ball rolling on tax reform Week ahead: Senators work toward deal to fix ObamaCare markets MORE embodies bold leadership and Iowa Republicans appreciate leaders who stand up for them and fight big government recklessness," A.J. Spiker, the chairman of the Iowa GOP, said in a statement. "Ted Cruz is, without a doubt, a leader Iowans will want to meet. As the 'First in the Nation' state, we look forward to introducing Senator Cruz to Iowa Republicans."

Last month, Cruz sought to downplay his interest in a presidential run, saying in a post to his Facebook page, his focus was helping the GOP retake the Senate.

"I was elected because thousands of grassroots conservatives came together to protect the Constitution, shrink the federal government, and promote growth and opportunity," Cruz said. "It is a continued source of amazement that the simple fact that I am working hard with like-minded Senators to keep my promise is seen as newsworthy and cause for wild speculation."