The Alabama operation is among the first examples to come to light of such underhanded methods on social media in American politics. But because such efforts are generally very easy to hide and very difficult to trace, it is possible that other instances have gone undetected.

In 2017, through a fund called Investing in Us, Mr. Hoffman gave money to a small company, American Engagement Technologies. The company’s leader, Mikey Dickerson, is a former Google employee who founded the United States Digital Service during the Obama administration to try to upgrade the federal government’s use of technology. An associate of Mr. Hoffman’s said that his total grant to American Engagement was $750,000, which also went toward other political races and social media research. Mr. Dickerson has declined to comment.

American Engagement then passed $100,000 to Mr. Morgan and other researchers. Mr. Hoffman’s statement says the money went to Mr. Morgan’s company, New Knowledge, but Mr. Morgan said the Alabama project was carried out separately from the company.

The revelations of and fallout from the operation are being closely watched. Some American political experts fear that the Russian influence campaign of 2016 could become a dangerous model for stateside politics, with dirty tricks on social media becoming common. Laws and regulations have not been updated to cope with such a threat.

On Saturday, Facebook shut down five accounts, belonging to Mr. Morgan and other unnamed individuals, in response to their use of “inauthentic” operations on the platform. Twitter declined to say whether it had taken any action in the Alabama case.

Senator Jones and Democratic Party officials have said they were unaware of the social media tactics and denounced them. Mr. Jones has called for investigations by the Justice Department and the Federal Election Commission — a call Mr. Hoffman joined on Wednesday.

“We cannot permit dishonest campaign tactics to go unchecked in our democracy — no matter which side they purportedly help,” Mr. Hoffman said in his statement.