Joe Walsh, a conservative radio show host and a former Tea Party congressman from Illinois who is expected to announce a long shot primary challenge to President Trump in the coming days, is wooing the husband of Kellyanne Conway, Mr. Trump’s former campaign manager, for a senior role in his campaign — a signal of his intent to troll Mr. Trump into engagement with him.

Mr. Walsh met on Thursday with George T. Conway III, the prominent conservative lawyer who is married to Ms. Conway, one of Mr. Trump’s longest-serving aides. During the hourslong conversation, the two discussed strategy going forward, according to two people familiar with the conversation, and Mr. Walsh asked Mr. Conway if he would join his nascent campaign in a leading formal role.

Mr. Conway, who has gained a large following on Twitter for his increasingly virulent critiques of Mr. Trump and his mental state but who has never worked on a political campaign before, did not commit to anything, telling Mr. Walsh he would think about it, the two people said.