Lori Lightfoot on Tuesday accused County Board President Toni Preckwinkle of trying to “bully” her out of the race for mayor and flatly declared that it will never happen.

“There’s a rumor floating around that I’m going to step down and take a deal … as some person in a Toni Preckwinkle administration,” Lightfoot said during a luncheon address to the City Club of Chicago.

“Let me be clear: That’s never happening. It was false the minute it was said. And what it shows, frankly, is the kind of machine-style bullying that has no place in the future of this city.”

Monica Trevino, a spokeswoman for Preckwinkle’s mayoral campaign, flatly denied the bullying charge.

“No one has offered Lori Lightfoot or anyone else a job in the administration,” Trevino said in an emailed statement.

“We have to first earn the support of Chicago’s voters with a strong campaign rooted in progressive values.”

Last month, Chicago Sun-Times columnist Michael Sneed wrote that Lightfoot “might entertain” exiting the race and endorsing Preckwinkle “in exchange for becoming corporation counsel in a Preckwinkle administration.” Sneed noted that the corporation counsel’s job would put Lightfoot, the former Police Board President who co-chaired the Task Force on Police Accountability, in a prime position to help a federal monitoring team implement police reform.

Lightfoot denied those rumors at the time. But the mere suggestion that she might cut a political deal caused a hiccup in her fundraising.

Instead of calling people to ask for their financial support, she had to first explain to them that she was in the race to stay.

That’s apparently why she came out swinging at the City Club and expanded on her “Boss Preckwinkle claims” after the speech was over while denying that a political deal was even offered.

“That’s wishful thinking on the part of people who, frankly fear my candidacy. I’m an independent person. I’m not gonna be beholden to anyone. And I don’t need somebody to hand me a job,” she said.

“The insulting part about that is that it suggests that I’m going to hand over my integrity to somebody like Toni Preckwinkle. My parents worked way too long and hard to provide each of us with the tools we needed to be successful. I’m not gonna retreat from any party boss who’s gonna try to bully me out of the race.”

It’s not the first time that Lightfoot has used Preckwinkle’s position as chairman of the Cook County Regular Democratic Organization to portray Preckwinkle as a political boss.

Earlier this month, Lightfoot used the party’s decision to honor outgoing Cook County Assessor Joe Berrios to lambast Preckwinkle.

Lightfoot branded Berrios the “poster child for bad government” for padding his payroll with relatives and “refusing to abide by any ethical standards,” in part, by thumbing his nose at the authority of the county’s inspector general and Ethics Board and fighting their restrictions at taxpayers’ expense.

She further accused Berrios of presiding over a flawed and unfair property assessment system that has favored the wealthy and politically-connected, at the expense of the average homeowner and the working poor.

The fact that Preckwinkle endorsed Berrios over Democratic primary winner Fritz Kaegi and leads a Democratic party that chose to honor him shows how “out of step” she is with everyday voters, Lightfoot said.

On Tuesday, Lightfoot blasted Preckwinkle for trying to “punish” Kaegi by leaving his name off the Democratic Party’s palm cards.

Party officials say Kaegi’s name was left off the palm card because he did not make the cash contribution that the party requires of candidates to be included in its campaign mailer.

“Whatever the price of admission is, this guy won an election fair and square,” Lightfoot said.

“The fact that they are still trying to seek retribution against him for taking on one of the most corrupt public officials of our time, Joe Berrios, and the fact that Toni Preckwinkle keeps wrapping her arms around him and protecting him really speaks volumes about where she is.”