In recent days, thousands of ballots have been rejected because voters did not meet a new requirement to provide an identification number inside the return envelope.

In Harris County, the state’s most populous county and home to Houston, election officials said Friday that 40 percent of roughly 3,600 returned ballots so far have lacked the identification number required under Senate Bill 1, as the new law is known. In Williamson County, a populous northern suburb of Austin, the rejection rate has been about 25 percent in the first few days that ballots have come in, the top election official there said.

“Twenty-five percent of mail ballots from the starting blocks is a big deal for our county,” said Chris Davis, Williamson County’s elections chief. “We’ve never seen it before. And yes, our hope is that we can get these voters to correct the defects in a timely fashion. But what if they don’t, because three months ago they didn’t have to? There’s a learning curve. There are going to be possibly painful lessons that their vote doesn’t count because they weren’t aware.”