The son of a former Virginia state senator has told federal investigators that U.S. Sen. Mark R. Warner discussed the possibility of several jobs, including a federal judgeship, for the senator’s daughter in an effort to dissuade him from quitting the evenly divided state Senate.

Warner was part of a string of high-powered Virginia Democrats who in early June pressed then-state senator Phillip P. Puckett not to go through with plans to give up his seat in the middle of a bitterly partisan battle over health care.

A Warner spokesman acknowledged Friday that the conversation occurred, but he emphasized that the senator had made no explicit job offer.

Puckett eventually resigned, throwing control of the chamber to the Republican Party and dooming Gov. Terry McAuliffe’s top legislative priority — expanding Medicaid under the Affordable Care Act. The incident also triggered a federal investigation into Puckett’s surprise exit, which came at a time when Republicans were planning to give Puckett a job on the state tobacco commission and confirm his daughter to assume a judgeship.

On June 6, three days before the state senator’s resignation became official, Warner called Puckett’s son, Joseph, and discussed an appointment to the federal bench as well as a potential corporate position for Martha Puckett Ketron, according to Joseph Puckett’s attorney, Charles E. “Chuck” James Jr. of Williams Mullen.

James said that Warner suggested a post for Ketron at CGI, at high-tech firm Warner helped lure to Southwest Virginia when he was governor a decade ago.

“My client, Joseph Puckett, received a phone call from Senator Warner, in which there was discussion of a CGI [ job] or a federal judgeship for the sister,” James said Friday.

Warner spokesman Kevin Hall said that the U.S. senator “brainstormed” with Joseph Puckett about potential private-sector jobs for Ketron. Hall also acknowledged that Warner discussed a judgeship, but he added: “He did not offer any job nor would he nor could he, frankly.”

Neither James nor Hall offered details about the conversation between Warner and Joseph Puckett.

“Senator Warner has been a close friend of Phil Puckett and his family for nearly 20 years,” Hall said. “Senator Warner called his son, Joseph Puckett, to find out what the situation was and brainstormed [about] whether Martha Ketron had considered other opportunities.

Senator Warner was not in a position to offer, and never did offer, any job to Mrs. Ketron. When he spoke to Senator Puckett the following day, it was clear that he had made up his mind and had already drafted his letter of resignation.”

The day after the conversation, Joseph Puckett recounted it in a text message to someone else.

“Talked to Warner for an hour last night,” Joseph Puckett wrote in the message, read to The Washington Post by two people close to the matter. He went on to report, “He had no answer for how Democrats could deliver for Martha except for a potential corporate gig of some kind, which is not what she wants.”

James would not confirm that his client had relayed the details of the conversation to federal investigators.

But two other people with knowledge of the investigation said that federal officials interviewed Joseph Puckett two weeks ago and that he told them about the conversation with Warner.

The text messages were also provided to investigators, the two people said.

Warner is seeking his second term in a race against Republican Ed Gillespie. The revelations come a week after The Post reported that McAuliffe’s chief of staff, Paul Reagan, suggested that Ketron could become the head of a state agency if Puckett stayed.

“If there’s something that we can do for her, I mean, you know, we have a couple of big agencies here that we still need agency heads,” Reagan said in a voice-mail message he left on Puckett’s phone. “We could potentially, potentially, subject to approval of the governor and so forth, you know, the Department of Mines, Minerals and Energy could be available.

“So we would be very eager to accommodate her, if, if that would be helpful in keeping you in the Senate. We, we would basically do anything.”

Reagan has apologized for efforts he described as “overzealous.” McAuliffe has said he was unaware of any offers made by Reagan, saying he was “disappointed” and characterizing his aide’s action as an “error in judgment.”

Ketron, who received her law degree from Campbell University in North Carolina in 2006, has done very little legal work in federal court and would not have been, at this point in her career, an obvious choice for a highly prestigious lifetime judicial appointment, associates said. At the time that Puckett was on the verge of leaving the Senate, there was an opening on the federal bench in Roanoke. Warner and U.S. Sen. Timothy M. Kaine (D) had recommended two others for the slot on June 6, earlier on the same day that Warner called Puckett.

They were Ward Armstrong, a former Democratic state delegate from Martinsville, and Elizabeth Dillon, a Salem lawyer.

President Obama ultimately nominated Dillon to the federal post.

Ketron was at the center of efforts to keep Puckett in the state Senate because he said he was resigning, at least in part, to clear the way for her appointment to a low-level state judgeship.

Circuit Court judges in Southwest Virginia had appointed Ketron to the bench as a Juvenile and Domestic Relations District Court judge on a temporary basis to fill a vacancy in 2013.

The state House of Delegates confirmed her for the post. But the state Senate has not approved her because Republican leaders in that chamber oppose putting relatives of sitting legislators on the bench under an anti-nepotism policy.

Puckett was also talking to leaders of the Republican-controlled state tobacco commission about a job for himself at the time of his resignation.

He has said there was no quid pro quo, but amid the uproar, he withdrew his name from consideration.

McAuliffe has acknowledged speaking to Puckett before his resignation but has denied discussing jobs with him. According to three people familiar with the matter who spoke on the condition of anonymity in order to discuss a sensitive matter, Warner, who has longer and deeper ties to Puckett and his southwest corner of the state, was asked to intervene. McAuliffe spokesman Brian Coy said the governor did not make that request.