"I support any plan that the chairman can get through his committee."
Rep. Charles Rangel (D-N.Y.) said the international headlines about Gray's case are "painful" and "embarrassing."
The legislation has been slowed by debate over dozens of amendments.
Sen. Mitch McConnell said the budget can help repeal the "failed policy" of ObamaCare.
House lawmakers are expected to tackle a spending bill after leadership pushed a vote originally expected on Wednesday.
Members debated appropriations amendments late into the night despite potential signs of trouble for passage of the year's first spending bill.
House GOP leadership denied a problem with the vote count.
But the Tennessee Republican said there's been discussion of the incident.
Sen. Ben Cardin warned the amendment was a "poison pill" for the legislation.
The provision would require certification that Iran supports Israel's right to exist.