HARTFORD — With the U.S. Senate returning next week after a summer break, Connecticut’s two senators said Thursday they will be fighting to extend the $600-per-week federal unemployment benefit enacted during the pandemic as well as obtaining more money for the state and local towns.

Sens. Richard Blumenthal and Chris Murphy will be building off a House-passed funding bill, but Republicans who control the Senate say the $3 trillion package is too expensive. The final compromise will determine how much money everyone receives — from public schools to nonprofits to individuals.

The $600 federal payment, which is in addition to state unemployment benefits, is scheduled to expire next week, but Blumenthal said the new payments could be made retroactive to cover any gaps for millions of workers unemployed nationwide. While a compromise could reduce the weekly payment, Blumenthal and Murphy said Senate Republicans are under pressure in an election year to offer additional benefits as the coronavirus spreads quickly in Republican-leaning “red” states across the South and the West.

“Very few of my colleagues want no extension,” Blumenthal told reporters. “The debate is about whether it should be $600 per week or some lesser amount. ... Let me just put it in very blunt political terms. November is just a few months away. They are going to face the wrath of voters. Those senators who face election this November will rightly be haunted if they fail to act. They will be haunted, and they will be given walking papers.”

Gov. Ned Lamont, who joined the senators at a news conference in Hartford, has favored a Republican plan for a one-time bonus of $450 as an incentive for those who return to work, saying last month that continuing the additional $600 per week might discourage employees from returning to work.

Lamont said Thursday that he did not oppose all extensions, adding, “No, it was specific to that amount of money, and I think as you’ve just heard they’re probably going to negotiate what that total amount of true-up is going to be.”