Kim's bid to remain as president comes in the face a rare public rebuke from the Bank's rank-and-file, who earlier this month declared that the institution faced a "crisis of leadership."
"The executive directors expressed unanimous support for an open, merit-based and transparent selection, with nominations open to all member countries," the executive board said in a statement.
Kim's five-year term ends on June 30.
Following an unwritten rule, the United States, the Bank's largest shareholder, has always chosen its president, while the International Monetary Fund leader has been drawn from Europe.
More From This Section
The executive board said Kim had been selected in 2012 according to new principles meant to ensure an "open and transparent" process and that the next president would be as well.
Kim was the first American candidate to face competition when Nigerian Finance Minister Ngozi Okonjo-Iweala also contended for the presidency.
The latest selection process begins at midnight on Thursday, launching a three-week window for nominations.
A medical doctor and former president of Dartmouth College, Kim was the first Bank president not to be drawn from the financial or political worlds.
After taking office in 2012, Kim announced a new goal of eradicating extreme poverty by 2030 and a sweeping internal reorganization to foster collaboration across the Bank.
However, the reforms appear to have alienated many staff members, who in employee surveys have reported feeling detached from senior management and unsure of the direction of policy.