The Benton County Sheriff is investigating threats made against Oregon Republican Party chairman Bill Currier, adding to the tension surrounding the standoff between the Democratic governor and runaway Republicans in the state Senate.

Mr. Currier and Becky Currier, his wife, received a total of three messages Sunday morning on their cellphones from what sounded like the same unidentified man threatening their family, according to audio recordings provided to The Washington Times.

“Bill, what if something bad happens to your family? What if somebody shoots them or something bad happens? I think you should be worried,” said one message.

Another message said: “F– you. What if something happens to your family? What if somebody gets hurt? What are you going to do then?”

The messages were received after Mr. Currier did interviews with news outlets, including Breitbart radio, defending the 11 Senate Republicans who left the state Thursday to deny a quorum to Senate Democrats seeking to pass a hotly contested cap-and-trade bill.

Solomon Yue, Republican National Committee member for Oregon, blamed Democratic Gov. Kate Brown for “creating a toxic political environment” by siccing state police on the Republicans, who fled the state and not disclosed their whereabouts.

“Now we see death threats against her political opponents,” Mr. Yue said in an email. “Oregonians must hold Gov. Brown accountable for any bodily harm to Chairman Currier and his wife. I call on Gov. Brown to publicly reject and condemn death threats against any Oregonian.”

Meanwhile, Democrats have knocked the GOP for the heightened tensions.

State Senate President Peter Courtney canceled Saturday’s session due to a “possible militia threat” from groups planning to rally at the state capitol, including the Oregon Three Percenters, which was involved in the 2016 armed occupation of the Malheur National Wildlife Refuge.

About two dozen protesters, mainly rural Oregonians, wound up attending the rally to support the Republicans, according to local news outlets.

“They’re standing up for us, and we need that,” a woman holding a sign with the message, “Stay Out Til 7-1-19/We’ve Got Your 6,” told Fox 12.

Republican state Sen. Brian Boquist responded to the threat of arrest last week by telling reporters, “This is what I told the superintendent: ‘Send bachelors and come heavily armed.’ I’m not going to be a political prisoner in the state of Oregon.”

The state legislature was back in session Monday but without the Senate Republicans, who stayed away for a fifth straight day.

In a Monday statement, state Senate Republican Leader Herman Baertschiger Jr. said that “no deal with the Democrats has been made,” despite rumors to the contrary.

The legislature must adjourn this Sunday, although the governor could call a special session to pursue the cap-and-trade bill, aimed at combating climate change by capping emissions and requiring larger companies to buy allowances for additional pollutants.

House Bill 2020 would bring the state an estimated $500 million in the first year, but foes, including loggers and truckers, argue that it would raise energy and fuel prices, forcing businesses to close or move out of state.

Mr. Baertschiger told radio host Lars Larson that he would be willing to return if Ms. Brown would agree to place the cap-and-trade measure on the ballot for the state’s voters to decide, but she has refused to do this.

“She knows that most voters would vote ‘no’ on this,” Mr. Yue said. “That’s the reason she won’t put it on the ballot.”

In an interview with The Associated Press on Monday, Ms. Brown said she would not negotiate with Mr. Baertschiger until he brings his Senate Republicans back into Oregon.

“If he wants to negotiate with the governor of the state of Oregon, he needs to be in the building,” she said. “Or at least be in the state of Oregon.”

Republicans pointed out that as a state legislator, Ms. Brown supported a 2001 walkout by House Democrats aimed at denying a quorum to Republicans over a redistricting bill, telling the AP that it was “very appropriate under the circumstances.”

Oregon would become the second state, behind California, with cap-and-trade legislation if the measure wins approval.

Democrats hold an 18-12 majority in the state Senate, but 20 votes are required to reach a quorum and conduct business.

The Democratic leadership has also said each of the wayward Republicans faces $500 per day fines for every day that the Senate fails to achieve a quorum.