After struggling for months to reach a deal with Democrats on how to broaden background checks, Mr. Coburn has written a 43-page draft of a bill that wouldn’t require private gun sellers to keep a paper record of the sale, according to a copy of the proposal.

Dropping the record-keeping requirement would make it difficult for Democrats to support the bill, according to individuals familiar with the negotiations. Democrats have argued there is no way to enforce the checks without some form of documentation…

However, the discussions between Mr. Coburn and Mr. Manchin have stalled, with neither side willing to yield on the record-keeping requirement, and Mr. Manchin started talks this week with Sen. Pat Toomey (R., Pa.), according to an aide familiar with the negotiations.

“The best hope we have right now of a bipartisan compromise is the discussion going on between Manchin and Toomey,” a Senate Democratic aide said Friday. “There is real potential there.”