CLEVELAND — Republicans are bracing for the possibility of a drawn-out fight over Donald J. Trump’s nomination that could stretch into the convention next week, a spectacle party leaders are desperate to avoid.

As the party’s governing body convened in Cleveland on Wednesday for a series of organizational meetings, the possibility that the smoldering dissent over Mr. Trump could become a full-blown eruption on the convention floor was a top concern.

Party officials were working in overdrive behind the scenes to beat back two challenges. One would force a vote that could theoretically open up the convention to any candidate who wants to seek the nomination. A second, organized by supporters of gay rights, would strike the entire Republican platform, which is staunchly conservative in its treatment of sexuality, gender and religion.

Supporters of the platform overhaul said the Trump campaign and the Republican National Committee were pressuring them to back down.