Mr. Pence must now balance his loyalty to an enraged president making baseless claims of voter fraud against his own political future and reputation. He also has to deal with how Mr. Trump’s talk of running for president again in 2024 could leave him with no lane to run in. It also makes it difficult for Mr. Pence to even start raising money if the president is floating his own name.

So far, Mr. Pence appears to be handling the pressure much as he has over the past four years: appearing to be unflinchingly loyal while also steering clear of engaging in Mr. Trump’s pressure campaigns…

Corey Lewandowski, a Trump ally, pushed for Mr. Pence to travel to the Pennsylvania Convention Center last week to observe votes being counted, according to a person familiar with the matter.

The image of the vice president — who has strong support with the social-conservative wing of the Republican Party — descending on a counting center in a state where Mr. Trump’s campaign was making baseless claims of widespread fraud would have aided the president’s narrative that the election was being stolen. Mr. Pence’s chief of staff, Marc Short, vetoed the request, the person said.