Updated 6:18 p.m. ET, 6/11/2014
(CNN) - Rep. Pete Sessions of Texas said he's planning to run for the House Majority Leader position, as Rep. Eric Cantor is expected to vacate the job on July 31.
Sessions, who chairs the House campaign arm known as the National Republican Congressional Committee, told reporters on Wednesday he will compete for the job.
5 aftershocks from Cantor's stunning upset loss
Following Cantor's stunning primary loss Tuesday, a senior GOP source tells CNN the Virginia congressman plans to relinquish his leadership position at the end of next month.
The House GOP conference will meet on June 19 behind closed doors and hold election for majority leader, at which time members will cast secret ballots.
While House GOP Whip McCarthy hasn't officially announced he's seeking the spot, many members believe he's the frontrunner.
Other names that have been floated as potential replacements include Louisiana Rep. Steve Scalise, Texas Rep. Jeb Hensarling, Ohio Rep. Jim Jordan and Georgia Rep. Tom Price.
Rep. Cathy McMorris Rodgers, who chairs the House Republican conference and is the top woman in GOP leadership, was also considered a possible contender, but she said Wednesday she won't seek the position.
"After much encouragement from my colleagues, conversations with my family, and many prayers, I have decided to remain Conference Chair at this time," the Washington state congresswoman said.