Senate Republicans are forging ahead with a two-step plan to end the record-breaking Department of Homeland Security (DHS) shutdown, but their House counterparts tell Fox News Digital they are not on board with the strategy.A swath of House Republicans have voiced growing frustration that a forthcoming GOP-only funding package does not include other policy priorities beyond funding immigration enforcement ahead of November’s midterm elections. President Donald Trump has set a June 1 deadline to fully fund immigration enforcement through a GOP-only bill, forcing Republicans to act quickly with little room for error.Before the DHS shutdown House Republican leadership teased a budget reconciliation sequel to the One Big Beautiful Bill Act that would incorporate a diverse set of priorities, such as a defense supplemental package, spending cuts targeting fraud and policies aimed at lowering the cost of living.Concerns among rank-and-file Republicans that a forthcoming budget bill will not include those provisions threaten to jeopardize that timeline.House conservatives have also fiercely objected to the Senate passing a bipartisan partial DHS bill carving out ICE and the Border Patrol from the normal appropriations process and keeping those two agencies unfunded.After Democrats in the upper chamber repeatedly filibustered DHS funding bills, the Senate approved legislation funding parts of the department that Democrats would support. "That sort of thing has never been done up here, to take an appropriations bill and sort of cherry pick what you don't want in it and isolate whole agencies … I’m against that whole premise."BEHIND THE SCENES OF CONGRESS' ELEVENTH-HOUR RUSH TO FUND THE DHSSenate Republicans are largely unified on keeping the package as narrow as possible out of concern that adding more to the pot could stall lawmakers’ progress.Senate Budget Committee Chairman Lindsey Graham, R-S.C., has sought to expedite the passage of a forthcoming budget bill by involving in the process just two panels — the Senate Judiciary and the Senate Homeland Security and Governmental Affairs Committees."The vast majority of Republicans stuck together to do something Democrats are refusing to do: Fully fund the Border Patrol and ICE for three and a half years through the Trump presidency," Graham said Thursday after the upper chamber adopted the budget blueprint.