Studies and reports have been obtained, arguments drafted and tea leaves obsessively examined — all in a bid to persuade Ms. MacDonough, who will determine the fate of several key liberal provisions, including a federal minimum wage increase Mr. Biden has championed. The decision on the wage increase could come as early as Wednesday…

The most anticipated ruling from Ms. MacDonough could come as early as Wednesday, lawmakers and aides said, on whether the Democratic proposal to gradually increase the federal minimum wage to $15 an hour is permissible under the budget rules. Under the “Byrd Rule,” established by former Senator Robert Byrd of West Virginia, a provision must affect federal spending or revenue to be included in a reconciliation measure.

With senators in both parties already mulling over possible amendments and changes to the $1.9 trillion relief proposal, which is set to pass the House at the end of the week, Ms. MacDonough’s determination could have significant consequences for the wage increase, as well as technical changes and potentially provisions related to pension funds and paid leave.