Mr. Murphy, in contrast, wants to raise taxes and increase spending on education and efforts to tackle the state’s opioid crisis, among other areas.

Mr. Sweeney also seems to have been put off by what he perceives as Mr. Murphy’s highhandedness when it comes to dealing with lawmakers.

Mr. Murphy drew the wrath of Mr. Sweeney and other legislative leaders, including Craig Coughlin, the Assembly speaker, over television ads financed by a political action group allied with the governor that have promoted a tax on the wealthy.

On social media, Senate Democrats have started their own counter campaign critical of the governor.

“Someone has told him that he’s the most powerful governor in the country,” Mr. Sweeney said in an interview. “But that also means the New Jersey speaker is the most powerful speaker in the nation. And it means the Senate presidency is the most powerful senate presidency in the nation. The three of us are partners, and maybe we’re not happy that we’re partners.”

He warned that lawmakers could override Mr. Murphy if he uses his line-item veto power — among the most powerful versions in the country — to reject specific spending proposals in the budget passed by the Legislature.

Mr. Murphy has indicated his lack of support for the budget and an override by members of his own party would be unusual and most likely further poison the dynamic among top Democrats.

The relationship between Mr. Murphy and the Legislature was strained from the start — many longtime members of Trenton’s political class were wary of an outsider governor who did not seem eager to work with them on their priorities.