Senate Majority Leader Mitch McConnell said Tuesday he hopes to approve further funding Thursday to buoy small businesses devastated by the coronavirus pandemic.

"I will work with Secretary Mnuchin and Leader Schumer and hope to approve further funding for the Paycheck Protection Program by unanimous consent or voice vote during the next scheduled Senate session on Thursday," he said in a statement.

Congress late last month approved a $350 billion small business loan program as part of a more than $2 trillion stimulus bill aimed at helping the economy recover from the coronavirus pandemic. The program is aimed at helping small businesses that provide the engine of employment and entrepreneurship in the U.S. economy.

The small business measures aim to help companies cover payroll and other expenses during the punishing outbreak. Firms with fewer than 500 employees can use the money to cover salary, wages and benefits, with a maximum loan of $10 million or 250% of monthly payroll

Still, the initial rollout of the ambitious program faced challenges as banks hurried to prepare themselves to assist with the efforts. In the run-up to the program's launch this past Friday, banks and trade groups pushed for more clarity around the program and sought to manage expectations about how quickly they could dole out the funds.

This week, there were technical glitches with the system amid the flurry of demand. Some banks are still awaiting guidance from the Treasury on promissory notes clarifying the terms under which the loans are valid, a person familiar with the situation told CNBC. The Treasury did not immediately respond to a request for comment.

Still, some banks viewed the announcement of more funds positively, believing it would reassure small business owners that the system would have enough money to cash their checks, even as it works through the technical issues, that person said.

Nonetheless, initial challenges may be a point of debate in any proposal to add to the program.

A spokesperson for Sen. Chuck Schumer said in a statement to CNBC, "Senator Schumer has not heard from Senator McConnell, and [the small business committee's ranking member] Senator Ben Cardin D-Md has not heard from Senator Rubio."

Sens. Cardin, Schumer and others issued a release Tuesday urging that a portion of the small business lending program be reserved for companies without relationships to banks. In an effort to expedite the process of delivering loans, some banks have hoped they can prioritize those with whom they have existing relationships.

But Schumer and other Democratic senators warned that some companies may get left behind.

"Without proactive and sustained outreach, we can expect that underserved communities will be disproportionately harmed – just as they were during the Great Recession when minority business enterprises suffered a precipitous decline in the proportion of SBA-backed loan approvals," they said.

Despite concerns about the existing program, Congress and the administration are under pressure to offer relief quickly. Administration officials and lawmakers have acknowledged the loan program may not be enough. Mnuchin has told CNBC he would ask Congress for more money if more is needed.

On Tuesday, the White House is expected to hold a virtual meeting with a number of major banking executives to discuss issues including small business support.

The Senate, meantime, had not been expected to be back in session before April 20. McConnell said Tuesday that a session was scheduled for Thursday.

While most senators are out of Washington, the chamber can pass legislation during regular pro forma sessions, brief periods of activity usually overseen by at most a few lawmakers. Any one senator can block the passage of a bill by unanimous consent.

After passing the Senate, the legislation would need to be approved by the Democrat-led House of Representatives. Speaker Pelosi told CNBC last week she that wants more small business loans as part of a bigger fourth financial relief package the Democrats have begun to discuss.