(L-R) Senate Majority Leader Mitch McConnell, Secretary of Treasury Steven Mnuchin, and Senate Minority Leader Chuck Schumer hold a meeting to discuss a potential economic bill in response to the coronavirus, COVID-19, in Washington, DC, on March 20, 2019.

Senate Democrats and Republicans are nearing a deal Sunday that could inject roughly $370 billion into loan programs for small businesses, a person familiar with the negotiations told CNBC.

The talks come after the $349 billion Paycheck Protection Program, which offers forgivable loans to small businesses, ran out of money on Thursday. Democrats rejected a proposal to refill the fund two weeks ago. They instead argued for changes, including adding more money to support federal testing, hospitals and local governments. They've also pushed to ensure groups without banking relationships get access to the program and SNAP benefits.

As of Sunday night, Democrats and Republicans were negotiating a deal that would allocate $310 billion more into the Paycheck Protection Program, setting aside $60 billion of that sum for rural and minority groups. Another $60 billion would go to the Economic Injury Disaster Loan program, a separate program offering loans for small businesses administered by the Small Business Association.

Senate Majority Leader Mitch McConnell, R-KY, hosted a call Sunday afternoon to provide an update on the package for members, along with President Donald Trump, Chief of Staff Mark Meadows and Treasury Secretary Steve Mnuchin, a senior Republican aide told CNBC.

In the call, Mnuchin said the program will not include the Democrats' push for state and local government funding, the senior Republican aide said. Democrats are still pushing for state and local funding, the person familiar with the negotiations said.

Mnuchin also said SNAP demands will not be in the package, the senior Republican aide said. He said discussions continue over federal funding for coronavirus testing, which he is discussing with McConnell, Sen. Lamar Alexander, R-TN, and Sen. Roy Blunt, R-MO.

Earlier in the day, Mnuchin said the deal could include $75 billion in funding for hospitals and $25 billion in funding for testing.

Rep. Steny Hoyer, D-MD, sent out a note Sunday evening saying "pending agreement" the House could take up the bill as soon as Wednesday.