In the poll, Lt. Governor David Dewhurst scored the highest with 38%. Ted Cruz came in second with 27% and Craig James and Tom Leppert both scored 7%. 19% of the poll's respondents stated they would like to see another Republican run in the race for U.S. Senate.

In a previous Texas Tribune poll from last October, Dewhurst scored 22% and Cruz 10%.

"David Dewhurst is roughly splitting the extremely conservative primary voters with Ted Cruz," said poll co-director Jim Henson, who teaches government at the University of Texas at Austin and runs the Texas Politics Project. "That's not a sign of failure or defeat, but it is a yellow flag."

During this current round of polling, the Texas Tribune also polled likely Democrat voters for the upcoming primary. Of the actual candidates named, Sean Hubbard scored the highest with 12% of the vote.  However, the majority of the Democrat poll's respondents, 55%, wanted another Democrat to jump into the race for Senate.

The Texas primaries are expected to take place in April or May of this year.