At least six Democrats are interested in succeeding Jennings as president pro tem of the Senate.
One of them, Sen. Mary Kay Papen of Las Cruces, said Thursday she is seeking the position because both Republicans and Democrats asked her to run.
Papen, 80, said in an interview that she prides herself on being fair and working with senators of both parties.
She said she wants the Senate to focus on doing what is right for New Mexico, not partisanship. But, Papen said, she would not be a pushover for anybody, including Republican Gov. Susana Martinez.
"I'll stand toe to toe with her if I have to," Papen said.
Another Democrat from southern New Mexico, Sen. John Arthur Smith of Deming, also could be in the mix for Senate president. Perhaps the most conservative Democrat in the Senate, Smith would likely pick up Republican supporters in a run for the body's presidency.
Smith, 71, said he did not want to talk about the leadership position yet. Papen and others said Smith may be happy chairing the Senate Finance Committee and not want to leave it.
Republicans, who will be outnumbered in the Senate 25-17 next year, say they want a president who will be fair in establishing the system for committee assignments. These are important because senators have to navigate their bills through committees to have a chance of their passing.
"John Arthur's great. Mary Kay's very fair," Neville said.
The other four Democrats known to be considering a run for the Senate presidency are Sens. Pete Campos of Las Vegas, Linda Lopez of Albuquerque, Carlos Cisneros of Questa and Phil Griego of San Jose.
Campos, 59, a community college president, has been in the Senate for 22 years. He said he was interested in replacing Jennings, but that this was only a discussion stage.
As the maneuverings continue, the selection could be settled by Democrats alone if they unite behind one candidate. But Republicans can have a voice in the selection if Democrats are divided between the contenders.
Jennings, of Roswell, retained the Senate presidency in 2008 by building a coalition of Republicans and Democrats. One of his rivals for the job was Cisneros, 64, who will try again.
Cisneros has been a senator since 1985. He will have the second-most seniority in the Senate come January. Only John Pinto, 87, a senator since 1977, will have served longer than Cisneros among sitting senators.
Lopez also wants the leadership job and, like Papen, she said she can work compatibly with Republicans.
Lopez, 48, chairs the Senate Rules Committee, a job she would give up if she becomes Senate president. Her committee does background checks on the governor's cabinet nominees as part of the Senate confirmation process.
At least a few Republicans have complained that the Rules Committee can be political.
For two years, it has not scheduled a confirmation hearing for Hanna Skandera, Martinez's designate as secretary of public education. Skandera is running the education department with full powers, even though she has not been before the Rules Committee.
Lopez asserted herself this fall by calling for the firing of Keith Gardner, Martinez's chief of staff.
Gardner, in a profanity-filled conversation secretly taped by a friend, said he hated Jennings and wanted to oust him from the Senate, even if it took $500,000 in campaign advertising.
Lopez said Gardner's language and conduct were unbecoming of someone receiving a paycheck from taxpayers. She called on Martinez to fire him. Gardner remains in the position. Still to be seen is whether his working relationship with Democrats in the Legislature has been damaged beyond repair.
Griego, 63, may be another senator who could pull Republican votes if he runs for president of the body.
He calls himself a rural senator. His district stretches 210 miles, from north of Santa Fe almost to Ruidoso. Martinez's camp helped Griego win his primary election against a more liberal Democrat from Santa Fe.