Senators Keith Ingram of West Memphis and Jimmy Hickey, Jr. of Texarkana, say they'll lead the effort to abolish the office when the legislature convenes in January.
Right now, the office is empty because the previously elected Lieutenant Governor resigned in the middle of his term.
Senator Hickey says getting rid of the office would save taxpayers about $450,000 a year.
To abolish the Lieutenant Governor's office requires voter approval of a constitutional amendment.
The legislature may refer up to three proposed amendments during each regular session.
"The office is a vestige of the early 19th century, before telephones were common and before computer technology was ever dreamed of," says Ingram.
"Under the Arkansas constitution, the Lieutenant Governor's only duty is to preside over the Senate and to serve as acting governor when the governor is out of state or unable to discharge the powers of the office."
Senators Ingram and Hickey plan to co-sponsor a Senate Joint Resolution, which if adopted by the legislature, would refer to the November, 2016, general election ballot a proposed constitutional amendment abolishing the office of Lieutenant Governor as of January 1, 2019.
There would be no statewide election for the office in 2018. The winner of this year's contested election for Lieutenant Governor still would be able to serve out the full four-year term to which he is elected in November.
"There are many pressing needs for the revenue now being spent on the office of Lieutenant Governor. People want us to streamline government, and this is a great way to do it," Ingram says.
Hickey says he and Ingram were announcing their plan well in advance of the legislative session so there would be sufficient opportunity to answer any questions the people have.
"I'm confident that when voters are informed about the issue they will be willing to abolish the office," Hickey said.
"I've gotten a lot of support for this proposal from people in both political parties, so I'm encouraged."
Both Lieutenant Governor candidates this year, Democrat John Burkhalter and Republican Tim Griffin say they will not support the effort to do away with the office.