A runoff in the Republican race to fill Kay Bailey Hutchison’s Senate seat may be inevitable. A new University of Texas/Texas Tribune poll shows Lieutenant Governor David Dewhurst has support from forty percent of likely voters and Ted Cruz, former Solicitor General and Tea Party darling, has 31 percent.

Former Dallas Mayor Tom Leppert had the support of seventeen percent of those polled and former professional football player and ESPN commentator Craig James had a paltry four percent. A runoff for the Republican primary would be held on July 31.

Dewhurst’s opponents have used the delayed primary—pushed March 6 to May 29—to gain ground, according to Jim Henson, poll co-director who heads the University of Texas at Austin’s Texas Politics Project. “Ted Cruz and Tom Leppert are claiming some real estate,” Henson told the Texas Tribune‘s Ross Ramsey. “The extra time played to the benefit of Cruz and Leppert.”

Dewhurst seemed to be showing some signs of strain in a radio interview with KPRC’s Matt Patrick last Wednesday. “I was surprised to hear Dewhurst as publicly frustrated and angry-sounding as he did on the radio yesterday afternoon,” broadcast journalist and political analyst Scott Braddock noted on his blog. Among Dewhurst’s choice quotes on air? “There’s isn’t anything moderate about me except I don’t scream at people” and “Everybody else is in diapers compared to me.”

Both campaigns have been drawing on the endorsements of higher profile politicians lately. Dewhurst’s campaign released a new thirty-second television spot Monday that features Governor Rick Perry saying he is “proud” to endorse the lite guv. “David Dewhurst’s the one candidate best prepared to make conservative change happen in Washington. Don’t let anyone tell you different,” Perry says in the ad. “Our country needs him, so does Texas. And, in recent days, some Texas voters have received robo-calls from Sarah Palin urging them to vote for Ted Cruz.

Meanwhile, a runoff also appears likely in the Democratic primary, which has been largely out of the spotlight: in that race, 29 percent of those polled said they supported former state representative Paul Sadler. But Sean Hubbard is nipping at Sadler’s heels with 25 percent of support.