Mr. Vance told reporters on Tuesday not long before he took the stage at a campaign event in Philadelphia — the same city where Ms. Harris will appear with Mr. Walz for the first time as the Democratic ticket — that he had left Mr. Walz a voice mail message congratulating him on being selected and saying he looked forward to a “robust conversation.”

“Maybe he’ll call me back,” Mr. Vance said. “And maybe he won’t.”

From the stage, Mr. Vance — a political acolyte of the Silicon Valley billionaire Peter Thiel — hammered Mr. Walz as “a San Francisco-style liberal,” suggesting that Ms. Harris’s selection was a result of lobbying from “the far left of her party.” When asked if he shared any common ground with Mr. Walz, his new rival for the vice presidency, Mr. Vance smiled.

“Well, look, I mean, yeah, we’re white guys from the Midwest,” Mr. Vance said. “I guess there are similarities there.”