Emails released Monday provide new examples of a Clinton Foundation official seeking access to the State Department on behalf of donors at a time when Hillary Clinton led the department.

The emails—obtained through a lawsuit by conservative watchdog group Judicial Watch—could fuel criticism that the Clinton family’s charitable foundation, in fundraising with wealthy donors, corporations and foreign nations, created a conflict of interest for Clinton during her work as the nation’s top diplomat.

In an exchange from June 2009, a Clinton Foundation official and longtime aide to former President Bill Clinton wrote to Huma Abedin, a top adviser to the secretary of state, seeking a meeting between the crown prince of Bahrain and Clinton.

“Cp of Bahrain in tomorrow to Friday. Asking to see her,” the official, Doug Band, wrote, using shorthand for “crown prince.” He added: “Good friend of ours.”

Abedin, a longtime confidante of Clinton who is now working for her campaign, responded that Crown Prince Salman bin Hamad al Khalifa had sought a meeting with Clinton the previous week “thru normal channels,” and that the secretary of state had said she “doesn’t want to commit to anything for Thurs or Fri until she knows how she will feel.”

Two days later, Abedin wrote to Band again. “We have reached out thru official channels,” she wrote.

The new emails were obtained through a Freedom of Information Act lawsuit filed by Judicial Watch against the State Department. The 725 pages of emails from Mrs. Clinton’s personal server included material that wasn’t handed over to the government as part of the Democratic nominee’s archive.

Josh Schwerin, a spokesman for the Clinton campaign, defended Clinton’s actions as secretary of state. “Once again this right-wing organization that has been going after the Clintons since the 1990s is distorting facts to make utterly false attacks,” he said. “No matter how this group tries to mischaracterize these documents, the fact remains that Hillary Clinton never took action as secretary of state because of donations to the Clinton Foundation.”

An expanded version of this report appears at WSJ.com.