Former Mississippi Governor Haley Barbour (R). (AP Photo/Bill Haber)

The intense clash between Sen. Thad Cochran (R-Miss.) and tea party challenger Chris McDaniel in Mississippi's Republican primary race has fueled another skirmish that is growing more and more personal: The fight between former Mississippi governor Haley Barbour (R) and the antitax Club for Growth.

Barbour backs Cochran, who was forced into a runoff against McDaniel in Tuesday's primary. The Club was one of McDaniel's earliest backers and among the biggest critics of Cochran. On Wednesday, the orginization called on Cochran to forfeit after he failed to clinch the nomination. That didn't sit well with Barbour.

"They’ve bled themselves white trying to get the only scalp left," he told The Washington Post on Wednesday. "They know if Cochran wins they’re out of business. They will have gone 0 for 2014."

The Club hit back on Thursday, calling Barbour "desperate."

"You’d be hard pressed to find a more desperate person than a Washington lobbyist who thinks his fees are being threatened," said Club spokesman Barney Keller in a statement. "Haley Barbour is supposed to be all powerful, but he’s losing a race in his own backyard, and his influence business in Washington will take a big hit when he no longer has a Mississippi senator in his back pocket."

The feud predates the Mississippi Senate race. As Robert Costa reported last year, Barbour encouraged party officials to tell donors not to give money to the group, which is known for backing insurgent challengers against more moderate Republican incumbents.

It's common for heated races to spur secondary quarrels between surrogates. But the McDaniel-Cochran showdown has grown nastier than most. So has the Barbour-Club fight.

With the June 24 runoff looming, the race just got another subplot.