Senator Tim Scott of South Carolina, one of the top contenders to become Donald J. Trump’s running mate, will host a gathering in Washington next month featuring Republican donors who so far remain publicly uncommitted to the party’s presidential ticket.

Pitched as a meeting of Great Opportunity Policy, a tax-exempt group that supports Mr. Scott’s political agenda, the private event on June 19 will double as a fund-raiser just as Mr. Trump’s vice-presidential search is expected to start heating up.