Existing duty suspensions on more than 600 products expired at the end of last year.
Earlier this week, the House seemed poised to pass the bill after top Democrats and Republicans on the Ways and Means Committee introduced the vetted legislation, which would have allowed components needed for products that are not made domestically to come into the country duty-free.
"In this new Congress, the House and Senate must work together on a clear path forward to get the MTB to the president’s desk as soon as possible," Dempsey said.
When the bill was introduced on Tuesday, Ways and Means Chairman Dave Camp (Mich.) said he was hoping to "move this much-needed relief as quickly as possible."
The measure is a top trade priority for Camp, according to a House aide.
The package includes provisions from more than 2,000 bills introduced in the House and Senate by about 170 lawmakers.
Meanwhile, the manufacturing sector showed signs of weakness in the last six months of the year.
A report on Wednesday showed that the manufacturing industry expanded in December after contracting the previous month, which was mostly due to concerns over the scheduled tax breaks and spending cuts.
Congress cleared the tax package measure for President Obama's signature on Tuesday and he signed it on Wednesday night.
An employment report on Thursday showed that manufacturers shed 11,000 in December amid an addition of 215,000 positions in the private sector, according to ADP.