More than 20 Republicans joined Democrats in voting to invoke cloture on the nomination. The final vote on Hochberg's confirmation is expected Wednesday afternoon.
The conservative Club for Growth has opposed the nomination because the group wants to wind down the bank's activities, arguing that the Senate should turn back Hochberg "until a real plan is implemented for reducing the Bank's authority with the ultimate goal of ending its charter completely."
Hochberg was easily confirmed in 2009 — by unanimous consent in the Senate — and saw the bank through a reauthorization fight last year that increased the total financing the bank can guarantee, to $140 billion from $100 billion.
The group argues that the increase in funding authority "means its ability to distort the free market by handing out huge discounted loans and loan guarantees to special interests has only grown in size."
The bank provides loan guarantees to foreign companies that want to do business with U.S. exporters. Last year, the bank provided more than $35.7 billion in financing.
Business groups broadly support the work of the bank and have pressed for Hochberg's confirmation.
Heritage Action said Wednesday in an emailed statement that it opposes the existence of the Bank because it "distorts markets by backing risky loans with taxpayer dollars."
"The Bank uses the guarantee of taxpayer money to back politically favored companies and projects, and makes a concerted effort to export the Obama administration’s global warming agenda," the group wrote.
"The Bank’s primary function is to encourage exports of U.S. products and services, but its contribution to total exports is negligible."
This story was updated at 12:45 p.m.