Gohmert has ruled out a bid against Cornyn, however.
Cornyn is not popular with some members of the GOP base, who have criticized his work while heading the National Republican Senatorial Committee and are unhappy that he's not backing an effort by Tea Party-affiliated Republican senators to use the looming debt-ceiling battle to try to force defunding ObamaCare. That effort has the support of Sen. Ted Cruz (R-Texas).
But Cornyn has long been preparing for a potential primary challenge. The Texas senator has nearly $6 million in the bank for his reelection effort, and recently brought on former FreedomWorks grassroots director Brendan Steinhauser to manage his campaign. Steinhauser was a major Cruz booster with close ties to Tea Party activists in the state.
This post was updated at 10:05 p.m.