Mr. White has also attacked the governor’s character, calling him a “career politician” who oversees “a political machine.” He has pointed out that Mr. Perry has been living in a $10,000-a-month apartment since the governor’s mansion was destroyed by an arsonist in 2008. Mr. White has needled Mr. Perry for the size and cost of his security detail.

Image Gov. Rick Perry of Texas is running for an unprecedented third full term. Credit... David J. Phillip/Associated Press

And Mr. White has also resurrected accusations that the governor has made some of his friends wealthy with state largess over his nine years in office, and has questioned Mr. Perry’s use of public funds to entice large corporations to Texas.

Mr. Perry has avoided committing to a debate, saying he will not agree to one until Mr. White releases his tax returns from the 1990s. “Bill White is hiding something on his tax returns,” he said Thursday.

Mr. White called the request absurd, pointing out that he has disclosed tax returns for all his years in elected office and his financial disclosure statements from his years as a deputy energy secretary in the Clinton administration. Mr. Perry, he asserts, is dodging a debate to avoid being held accountable for the state’s budget woes, among other things.

The debate over the debate ended in a stalemate this week, and it looks unlikely that a debate will ever be held, both sides now say.

Whether Mr. White’s punches are landing is hard to gauge, political strategists and pollsters say. Some Republican strategists say the race has tightened only because Mr. White’s campaign, and a Democratic group called Back to Basics, have been advertising since July, often with negative messages, while Mr. Perry put his first television advertisements on the air this week.