The U.S. House is gearing up to pass the $1.9 trillion COVID relief package later this week, sending the massive bill to the evenly divided Senate. Once it arrives in the upper chamber, Senate Finance Committee Chairman Ron Wyden (D., Ore.) says he'll push for an additional month of enhanced unemployment benefits.

The House bill contains an extra $400 per week in jobless aid through the end of August — but Wyden wants to see those benefits to last through the end of September, as President Biden originally outlined in his American Rescue Plan.

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Lawmakers are rushing to pass the bill before the current $300 weekly boost expires in mid-March. The original $600 per week in enhanced benefits lapsed at the end of the summer last year, when lawmakers couldn't come to an agreement on another relief bill until December.

Wyden's counterpart in the House, Ways and Means Committee Chairman Richard Neal (D., Mass.), told Yahoo Finance he's open to the idea, but it's not clear if it's doable at this point.

"We have to adhere to the rules of reconciliation," said Neal. "I'm always open to the idea of improved legislation, but I think that what we did in the plan that came from the committee is pretty remarkable."

Under the rules for budget reconciliation, the process Democrats are using to pass the relief package without Republican support, the bill's price tag can't exceed $1.9 trillion.

'A prescription for trouble'

When asked by a reporter on the call what he would take out of the bill to make room for the jobless benefits, Wyden declined to give specifics. He reiterated his point that letting the benefits expire at the end of August — when Congress is typically out of session and lawmakers are in their home states and districts — is a "prescription for trouble."

Neal called Wyden's concerns about timing "reasonable" and should be addressed if possible, but he's hopeful the economy will be in much better shape by the time the benefits are set to lapse.

“I’m optimistic that by the end of August, we will be in a place where these enhanced benefits won’t be needed, but we will continue monitoring and revisit if that’s what’s needed at the time," Neal said in a statement to Yahoo Finance.

While Wyden is pushing for an extra month in the relief bill, his ultimate goal would be to establish automatic triggers that would tie the enhanced benefits to unemployment. As the economy recovers, the weekly benefits boost would phase out. Wyden says this move would give unemployed workers security, end so-called "benefits cliffs" and take politics out of the decision-making process.