As Congress enters a do-or-die stretch on efforts to pass a cryptocurrency regulatory overhaul, federal enforcers are signaling they’re poised to dial up oversight if lawmakers stumble.

Senators adjourned for their August recess without reaching a compromise on a landmark proposal known as the Clarity Act (H.R. 3633) that would give the Securities and Exchange Commission and the Commodity Futures Trading Commission fresh power to regulate digital assets.

Majority Leader John Thune (R-S.D.) teed up a vote to proceed to the measure when the chamber returns in mid-September. But with time dwindling to strike a deal ahead of …