McConnell said he hopes Sens. Orrin Hatch (R-Utah) and Ron Wyden (D-Ore.) will continue to try to find a way forward on amendments to the legislation, after talks appeared to hit a hurdle Tuesday.
"It is my hope that we would be able to process a number of amendments … and then move forward, and we'll have a couple of days to accomplish that," he said.
Senate Democrats slammed McConnell on Tuesday, suggesting the Kentucky Republican is breaking his promise to have an "open" amendment process for a key piece of trade legislation.
Sen. Bob Casey (D-Pa.) said while the Senate has been debating the "fast-track" trade legislation, which will allow the president to get his trade deals approved by Congress with a simple majority vote, there have been "very few votes and very few amendments."
"To shut it off and to stop at this moment in time, as some seem to want to do, I think is contrary not just to what the Senate should do on something as substantial as trade promotion authority," he said.
McConnell tried to counter some of the criticism over his procedural move, saying that it's "not the end of the story" on the trade legislation.
But he suggested that if Democrats want additional amendment votes, they will have to drop their objections and cooperate, adding that "just a little cooperation" is needed from Democrats.
More than 100 amendments from both sides of the aisle have been filed to the trade legislation. But so far, senators have had just two votes on amendments, both earlier this week.
Sen. Barbara Boxer (D-Calif.) suggested that unless senators can get more amendment votes, including on those she submitted, she would try to slow-walk the bill.
"I'm going to do everything in my power to stall this. I'm going to use every tool in my disposal," she warned.
The Democrats' criticism follows remarks from Senate Minority Leader Harry Reid earlier Tuesday. The Nevada Democrat said McConnell would file cloture on trade promotion authority Tuesday evening.
Sen. Jeff Merkley (D-Ore.) urged McConnell to allow more debate on the TPA, adding he should "keep his vision that he laid out on this floor that this would be an open process and a robust process."
Hatch tried to offer votes Tuesday evening on additional amendments, but Democrats objected to his request after Hatch declined to include a measure blocking McConnell filing cloture to end debate on the trade legislation until Wednesday.
Sen. Sherrod Brown (D-Ohio) said that filing cloture Tuesday "really does stifle the process."
Under Senate rules, lawmakers can take a procedural vote on the trade bill Thursday.
A war of words over the amendment fight played out between top Hill staffers on the sidelines of the floor fight.
Don Stewart, a spokesperson for McConnell, tweeted earlier Tuesday that staffers were trying to set up amendment votes, but Democrats were objecting to votes.
But Adam Jentleson, a spokesman for Reid, fired back that Republicans were only offering "2-4 amendment votes then shut down debate."
A key part of the debate on amendments focused on trying to get a vote on Sen. Maria Cantwell's (D-Wash.) proposal to reauthorize the Export-Import Bank.
Cantwell said she would continue to object to the trade bill until she got an answer on how the bank's life would be extended.
"Unti our colleagues give us an answer about something we've been clear about for more than a year, we are going to continue to object," she said. "We're not going to let this bank expire, the credit agency, without a fight."
McConnell said earlier in the day that the Export-Import bank would get a vote, but that it shouldn't be a part of the trade bill.
But Sen. Lindsey Graham (R-S.C.) warned his party's leader later Tuesday that he wouldn't vote with McConnell "on anything else this year" until he gets a vote on the Export-Import Bank.
"I'm tired of talking," he said. "There's over 60 votes in this body [for Export-Import.]"
- Updated at 7:45 p.m.