Rahm Emanuel and Bruce Rauner just can’t seem to keep the bromance alive.

The Democratic mayor started the day Friday by offering his Republican friend some pointed advice:

“Stop name-calling and just do your job.”

The governor responded by promising to send dead fish to Emanuel.

“I bought a gift for a special person,” Rauner said, laughing. “He’ll appreciate the gift as only the mayor can.”

They were once so close they took family vacations together and enjoyed expensive bottles of wine.

But now it’s four months into the fiscal year, the state of Illinois still doesn’t have a budget, and the city of Chicago is begging for help from Springfield even after the passage of an historic tax hike this week.

And the bromance could be on the rocks.

The latest flare-up actually began Thursday when Rauner’s office said the mayor needs to get serious about reforms or be “just another tax-and-spend politician who wants to blame someone else for their failures.”

Emanuel wasted no time responding.

“What Springfield needs to do is not call names to anybody. You’re 120 days and counting behind schedule, six thousand-plus kids have been thrown out of day care,” the mayor said at an unrelated event Friday morning.

“I would just say this to the governor and the governor’s office: You’re 120 days behind budget, six billion dollars and counting in not paying bills. Stop name-calling and just do your job.”

Mayor Rahm Emanuel speaks to the media at the opening of the New City Plaza in Chicago on Friday. | Kevin Tanaka/For the Sun Times

Mayor Rahm Emanuel speaks to the media at the opening of the New City Plaza in Chicago on Friday. | Kevin Tanaka/For the Sun Times

A couple hours later Rauner mocked the mayor during a news conference at a North Side butcher shop.

At the Paulina Meat Market, the governor bought two pork chops, a cut of beef tenderloin and two pieces of tuna, promising to send the dead fish to Emanuel.

Emanuel is notorious for years ago mailing a dead fish to a pollster whom he believed had screwed up —a veiled nod to a scene from “The Godfather.”

Despite his chuckling over the tuna steaks, Rauner said the budget stalemate is “extremely serious. “ He accused Democratic legislators of letting funding for day care providers lapse as a way to “pressure” the governor to buckle.

Rauner then ripped the mayor again for not doing more to get to the root of the city’s financial problems.

“He’s put in the biggest tax hike in Chicago history with no structural reform,” Rauner said. “I understand why he’s sensitive right now. He’s got a lot of really angry constituents, and they’re angry for a good reason. I am taking a position for the long term.”

Rauner held the event to announce that he had signed a bill allowing the Paulina Meat Market to get a liquor license. The shop needed both city and state approval because of an obscure law restricting liquor sales within 100 feet of a nearby church.

Rauner held up a copy of the legislation for the cameras, decrying it as an example of government overreach and red tape that the mayor should join him in eliminating.

“We need less bureaucracy, and we need local control,” the governor said. “That’s the key to our prosperity, and this issue summarizes it.”

The chief sponsor of the bill — the area’s state representative, Ann Williams, a Democrat — did not attend the news conference, though Rauner said he had worked with her to streamline the regulations. She did not return a call or e-mail for comment.

Rauner and Emanuel once worked on a lucrative business deal when they both worked in the private sector, and on other occasions they vacationed together. But for months the two have been battling over how to cope with the city’s budget pressures and pension obligations.

While the mayor has asked for help from Springfield, Rauner has insisted on “structural reforms” including restrictions on union collective bargaining and worker’s compensation.

The latest barbs came after the Chicago City Council signed off this week on Emanuel’s 2016 budget, which includes a $543 million tax hike to pay for growing police and fire pension payments. But the mayor’s plan still banks on Rauner signing legislation — already approved by the Illinois House and Senate — that would give Chicago 15 more years to ramp up to 90 percent funding level for the pension funds.

Emanuel also wants Rauner to back his plan to double the property tax’s homeowners exemption — from $7,000 to $14,000 — to cushion the blow for homes worth less than $250,000.

The homeowner exemption was passed by the City Council after the budget vote. But it still must pass in Springfield, where a House committee has already approved it.

On Thursday, Rauner spokesman Mike Schrimpf issued a statement blasting Emanuel’s budget. “It’s clear that less than 24 hours after passing the largest property tax hike in city history, the mayor is already laying the ground work for another tax hike because he is refusing to engage in passing structural reforms that will save Chicago taxpayers hundreds of millions of dollars. The mayor needs to get serious about whether he is going to be a reformer or just another tax-and-spend politician who wants to blame someone else for their failures.”

Rauner signaled again Friday that his old friend shouldn’t count on a bailout from Springfield. “I want Chicago to solve its problems,” the governor said.

When a reporter told Rauner that Emanuel said he’s not doing his job, the governor laughed.

He said he speaks with Emanuel every few days and would continue to do so. Still, “Let’s be candid about what’s going on here. There’s some hiding, dodging.”