The College Democrats of New Jersey were preparing to make an endorsement in one of the country’s most closely watched U.S. Senate primaries when calls began to come in from someone in touch with the campaign of Tammy Murphy, the presumptive front-runner and the wife of the state’s governor.

The caller, a female college student who works as a youth coordinator for the Democratic State Committee, wanted to know what Ms. Murphy’s campaign could do to block the group from endorsing Ms. Murphy’s main rival, Representative Andy Kim.

“Would a call from Tammy help?” the woman said she asked, while indicating she was relaying a message from the Murphy campaign.

Then, in a series of calls over the next two hours, the pressure from the caller, Keely Magee, escalated to warnings — about funding and future job prospects for leaders of the College Democrats, according to several people involved in the discussions and a recording of one call.