ALBANY — Rattling sabers amid a standoff, Democratic Party officials presented a pair of ultimatums to two warring factions in the New York State Senate if a reconciliation is not achieved in 2018, along with an outline of a possible deal to end the long-running feud.

The warnings, and hint of a possible rapprochement, came as a result of a letter from Byron Brown, the Buffalo mayor who is also chairman of the New York State Democratic Party, and several other high-ranking party officials. The letter was addressed to the heads of two Democratic clans: Senator Andrea Stewart-Cousins, who leads a group of 23 mainstream Democrats, and Senator Jeffrey Klein, who guides the Independent Democratic Conference, a renegade group of eight Democrats who collaborate with Republicans in Albany’s upper chamber.

The I.D.C.’s arrangement with the G.O.P. — which has afforded Mr. Klein and his cohort perks like committee chairmanships and bigger staffs — has been a constant source of party friction during the nearly seven years under Gov. Andrew M. Cuomo, a Democrat who has been blamed by some critics for not forging a deal between the two groups. That criticism has markedly increased since the election of President Trump, a development that awakened grass-roots progressive groups in the state to the presence of the I.D.C., and galvanized calls for unity, something Mr. Cuomo has said he too stands for.

But on Monday, the state party — which Mr. Cuomo controls — simultaneously offered an olive branch and threw down a gauntlet, telling Mr. Klein and Ms. Stewart-Cousins that the price for continued bickering could come via primary challenges (for the I.D.C.) or leadership changes (for the mainstream Democrats).