Sen. Lindsey GrahamLindsey Olin GrahamGOP and Dems bitterly divided by immigration We are running out of time to protect Dreamers US trade deficit rises on record imports from China MORE (R-S.C.) has a huge lead and is on the verge of avoiding a runoff in next Tuesday's primary, according to a new poll.

Graham has 49 percent support in a new poll conducted by Clemson University, just shy of the 50 percent he'll need to avoid a runoff. South Carolina state Sen. Lee Bright (R) is at 9 percent support, while none of Graham's other opponents crack 5 percent.

The two-term senator has essentially crafted the template for quashing a Tea Party challenge in his race, and his hard work looks like it will pay dividends next week. The 35 percent of undecided voters in the poll indicate he's likely to finish comfortably above 50 percent in the primary.

The live-caller Clemson University Palmetto Poll surveyed 400 frequent Republican voters from May 22-29 and has a margin of error of plus or minus 6 percentage points.