The behind-the-scenes competition between two of Iran’s most prominent conservatives ahead of the country’s presidential elections is intensifying. Although neither have declared an intention to run, Parliament Speaker Mohammad Baqer Qalibaf appears to be trying to prevent hardliner favorite Chief Justice Ebrahim Raisi from entering the race.

While the two have always sought to portray their relationship as cordial, there are indications that they are embroiled in a fierce competition to win conservative backing ahead of the June 18 vote...