Facing criticism for some of the donations given to his family’s philanthropy, Bill Clinton said on Thursday that the Clinton Foundation would no longer accept foreign or corporate money and that he would resign from its board should Hillary Clinton win the presidency.

Mr. Clinton’s announcement, which he relayed to foundation employees in a meeting on Thursday, followed the recent release of State Department emails mentioning donors to the Bill, Hillary & Chelsea Clinton Foundation who had contact with aides to Mrs. Clinton while she was the secretary of state.

The donations have become a lightning rod in Mrs. Clinton’s campaign against Donald J. Trump, who has repeatedly implied that foreign donors had corrupted his opponent’s tenure as secretary of state. On Tuesday, the Trump campaign pointed to an editorial in The Boston Globe titled “Clinton Foundation Should Stop Accepting Funds.”

The moves come amid concern among some Clinton allies that additional details could emerge about relationships between Mrs. Clinton’s State Department and foundation donors.