The Senate leader said he hopes for a "thorough, thoughtful and respectful debate."
"With cooperation, we can pass the bipartisan bill this week," the Republican leader said.
Republican senators are tiptoeing around the possibility.
A first vote on the long-stalled bill could come on Wednesday.
Two Democrats voted to move forward with the bill, which stalled in a 53-46 vote.
An article raised questions about Graham's military service.
The Senate will take a vote Monday to move forward with legislation to cut off federal funding.
Manchin is the first Senate Democrat to back the legislation.
"It's time they are investigated," Sen. James Inhofe said.
Nearly every year, a controversy erupts that shakes the political landscape.