A new poll out Tuesday shows a slew of potential GOP Senate contenders leading several rumored Democratic opponents.  

Lt. Gov. David Dewhurst (R) holds double-digit leads over former Rep. Chet Edwards (D-Texas), former state Comptroller John Sharp (D) and San Antonio Mayor Julian Castro (D), according to numbers from Democratic-leaning Public Policy Polling.

Republicans are already scrambling to position themselves for the open-seat Senate contest — a rare opportunity in Texas politics. Sen. Kay Bailey Hutchison (R-Texas) announced last week that she would not run for another term next year. 

Dewhurst, whom GOP insiders in the state say would start the race as the favorite in a GOP primary, leads Edwards 50 percent to 31; Sharp, 49 percent to 31; and Castro, 53 percent to 25. 

Dewhurst is the only rumored Republican candidate who reaches above 50 percent support in hypothetical general-election match-ups. 

Other potential GOP contenders, including Railroad Commissioners Michael Williams and Elizabeth Ames Jones and Dallas Mayor Tom Leppert, also top every rumored Democratic hopeful, but none of them tops 50 percent. 

The poll did not test former Texas Secretary of State Roger Williams (R), who was endorsed Monday by former President George H.W. Bush, or former Solicitor General Ted Cruz (R), another rumored contender. 

The early numbers are largely a reflection of name recognition, with Dewhurst being the best-known among the group. 

The poll found just 38 percent of voters have no opinion of Dewhurst. That number is significantly higher for other rumored contenders, ranging from 54 to as high as 72 percent.

The poll surveyed 829 Texas voters and has a margin of error of plus or minus 3.3 percentage points.