FRANKFORT, Ky. — Days after Matt Bevin took office in December, the Republican governor went to work on an ambitious project: persuading some Democratic state representatives from conservative districts to switch parties, giving Republicans control of the state House for the first time since 1920.

Two representatives did switch parties, while two more resigned to take other jobs. But Russ Meyer, a Democrat from Nicholasville, declined. Democrats held onto their majority, setting up a series of pivotal elections this November as all 100 seats are on the ballot.

Now, Bevin and Meyer’s discussion — chronicled in part by a voicemail the governor left on Meyer’s cellphone —has prompted the Kentucky House speaker to threaten Bevin with impeachment and has only increased the pressure in elections that could determine which party will control the last legislative chamber in the South still held by Democrats.

Tuesday, Meyer released a 42-second voicemail message he says Bevin left him the day after Meyer told Bevin’s office he would not be switching parties. Meyer, who is in his first term and is being challenged by Republican Robert Gullette III in November, called it proof Bevin threatened to use his office to punish him and his district if he refused to become a Republican.

“I want to make sure you understand where things are in my mind and the decisions that I’m going to make in the days ahead, weeks ahead, months ahead,” Bevin said, according to a recording where he identified himself by name. “I want you to be very aware of what the impact of those decisions will be as it relates to you, your seat, your district, etc., just so we have all of the cards on the table.”

Bevin spokeswoman Jessica Ditto confirmed the message was from Bevin. In a news release, she did not explain what Bevin meant by “the impact of those decisions” and how they would affect Meyer and his district. She called Bevin’s message “polite and personal” and said the fact that Meyer would wait nine months to release it is “proof of his continued insecurity about keeping his seat.”

“His desperate and partisan effort to misconstrue the conversations that he initiated is a discredit to the office he holds,” she said.

Bevin has been aggressive both in his courtship of and conflict with Democrats since he took office. Democratic House Speaker Greg Stumbo has already taken him to court over his vetoes of the state budget, and Democratic Attorney General Andy Beshear has sued Bevin three times. In July, Bevin angered Democrats again by approving a contract of up to $500,000 for a private law firm to investigate former Democratic Gov. Steve Beshear, who has denied any wrongdoing and accused Bevin of political gamesmanship.

Meyer said Blake Brickman, Bevin’s chief of staff, threatened road projects in his district as retaliation for not switching parties. Meyer said he decided to release the voicemail after the Bevin administration delayed a road project in his district earlier this month, which was first reported by The Courier Journal.

“I felt like, man, this is not right,” he said.

In a news release, Brickman denied he threatened road projects in Meyer’s district. Transportation Cabinet officials say construction was delayed because Steve Beshear approved the project without securing the land. Beshear has blamed Bevin for stopping the work. The state had to pay a $625,000 fine to the contractor as a result of the delay.

On Tuesday, Stumbo again threatened to impeach Bevin over the incident and called for a state and federal investigation. Bevin’s spokeswoman called that “ridiculous” and said “it is difficult for intelligent people to take him seriously.”

John Roach, a former state Supreme Court justice who worked on Bevin’s transition team, said he met with Meyer several times about switching parties, saying Meyer even inquired about resigning his House seat to take a job in the Bevin administration. Roach said none of their conversations included threats.

“My discussions with him were about how this would be good for Kentucky,” Roach said, adding he was disappointed by Meyer’s characterization of Bevin. “I mean, I’m sickened by it. It saddens me.”

Meyer confirmed he discussed taking a job in the Bevin administration, but said he decided against it.