Sen. Sherrod BrownSherrod BrownMajor progressive group unveils first 2018 Senate endorsements Congressional leaders unite to protect consumers Mnuchin weathers stormy confirmation hearing MORE (D-Ohio) said he lacks the votes right now to advance his amendment limiting the size of banks.

Brown held out hope that the measure, which he's offered along with Sen. Ted Kaufman (D-Del.), could win enough support to pass as the Senate debate moves forward.

"I don't think we do yet," Brown told Bloomberg's "Political Capital" when asked if he has the votes. "I think a week ago we weren't even close. I think this weekend we're closer."

The Senate will take up a variety of amendments starting this week on amendments to Democrats' Wall Street reform bill. The amendment that's been offered by Brown and Kaufman would place hard caps on the size of banks and the extent to which they can use leverage.

Other senators have been more reluctant to sign onto such language. Sen. Mark WarnerMark WarnerGOP, Dems hear different things from Trump Decaying DC bridge puts spotlight on Trump plan Overnight Cybersecurity: Dems split on Manning decision | Assange looking to make deal MORE (D-Va.), for instance, said this weekend on C-SPAN that he believes the size of firms isn't what matters, but rather, how well-regulated they are.

Brown and Kaufman's effort is aimed at quelling concerns over firms that are "too big to fail," or holding so many assets and complex contracts that their failure would have a cascading effect throughout the financial industry.

Still, Brown held out hope that more aggressive messaging would help win his colleagues over when it comes to backing their amendment.

"I think that by next week, as people have watched the Goldman hearings, as people hear these statistics -- like 63 percent of GDP, their combined assets -- as people hear the debate on this, I think senators in both parties increasingly are going to join Sen. Kaufman and me on moving forward with this," he said.