Mr. Trump has told advisers the issue will help the party win control of Congress next year and win back the White House in 2024. He has privately floated the possibility of an early presidential campaign announcement to underscore the message to conservative voters.

Many Republican candidates have fallen in line. Some have refused to concede defeats from 2020—and, like Mr. Trump, used fraud claims to raise money. Others seeking office have tailored their campaign messages to echo Mr. Trump’s claim that he won to avoid facing a backlash from his supporters…

This past year, Mr. Trump has supercharged the trend. In private conversations, he has said Mr. Gore was wrong to concede the race in 2000, people familiar with the remarks said. He has also told allies that his polling shows voter fraud motivates his base and argued that an announcement sometime next year that he would run for president would boost Republican turnout for the midterms, people familiar with the conversations said.

Fellow Republicans have dissuaded him from that strategy for now, the people said. But the disagreement was over whether Mr. Trump’s presence in the midterms would be a better motivator for Democratic voters than Republicans—not over the use of voter fraud as an issue, they said.