The Public Policy Polling survey of Republican primary voters found Dewhurst drawing 38 percent of the vote, followed by Cruz at 26 percent, former Dallas Mayor Tom Leppert at 8 percent and former ESPN analyst Craig James at 7 percent. Eighteen percent remained undecided.
A January PPP poll found Dewhurst leading Cruz 36 percent to 18 percent among Republican primary voters.
While the new poll shows Cruz gaining ground in the last three months, it does not suggest Dewhurst is any closer to drawing 50 percent of the primary vote, which he would need to avoid a runoff.
PPP director Tom Jensen wrote on the firm's website that Cruz appears to have momentum heading into the primary.
"Among voters who are familiar with Cruz — whether they have a positive opinion of him or not — Cruz leads Dewhurst 39-34," Jensen wrote. "That suggests that as Cruz's profile continues to increase, this race may continue to get closer."
The latest poll was conducted April 19-22 and has a margin or error of 4.9 percentage points.