High-resolution images show that Biden was adjusting the crease on his shirt and was wearing a rosary on his wrist. The claim is false.
In the video(34:37 minute), a mark on his shirt appears to be a crease, not a wire, which later disappears after a few seconds when Biden moves it.
Washington Reporter Matt Viser tweeted a statement by Kate Bedingfield, Deputy Campaign Manager and Communications Director for Joe Biden. Bedingfield denied the allegation.
In 2017 on Megyn Kelly Today show Joe Biden had revealed that he wears his late son Beau's rosary beads around his wrist. He had also said that when he runs for the 2020 presidential election, he would remember his son this way.
Therefore this claim is false. Democratic Presidential Candidate Joe Biden was not wearing a wire in his debate with Republican candidate Donald Trump.