Republicans had the sounded alarm after a series of polls showed a tighter race between Cruz and O'Rourke, a well-funded challenger who aims to become the first Democrat to represent Texas in the Senate in a quarter of a century. The latest poll has a margin of error of plus or minus 4.1 percentage points.
An O'Rourke victory would be a major coup for the Democratic Party as it tries to navigate a brutal map this year to flip two GOP-held seats and take a majority in the upper chamber.
A Quinnipiac poll of Texas voters last month showed a 6 percentage point lead for Cruz.
In Tuesday's survey, 93 percent of likely voters said they had made up their minds. Both candidates had the support of 94 percent of likely voters in their parties, while O'Rourke held a slim lead among independents.