WASHINGTON — Jeb Bush and Mitt Romney are scheduled to meet privately this week in Utah, raising the possibility that the two former governors will find a way to avoid competing presidential campaigns that would split the Republican establishment next year, two prominent party members said Wednesday night.

The meeting was planned before Mr. Romney’s surprise announcement two weeks ago to donors in New York that he was considering a third run for the White House.

Mr. Bush proposed the meeting, according to one of the party members familiar with the planning, who did not want to be quoted by name in discussing a secret meeting.

The original idea was for Mr. Bush, who announced his presidential ambitions in December, to show his respect for Mr. Romney, the Republican Party’s 2012 nominee. The meeting stayed on both men’s calendars even as Mr. Romney took steps to test the presidential waters, moves that could make the meeting awkward.