The deal would have allowed the bill's approval by unanimous consent, as its sponsor, Collins had advocated.
In exchange, Bond, who placed a hold on the bill in August, would be able to offer an amendment on the floor that would require the Postal Service to ensure that rates set for individual products -- including first-class mail -- are "fair and equitable," a Senate aide said. A similar amendment is in the House version of the bill.
Bond, who had previously said he wanted to give the full Senate a chance to consider his language, declined the offer, an aide said.
A spokesman for Bond did not confirm the move, but said, "There were some discussions, but until we know that ordinary Americans are going to be protected from unfair rate increases, Sen. Bond's position remains unchanged."
A Senate aide said Collins was willing to accept Frist's offer. Collins' spokeswoman said if the legislation "is killed in the U.S. Senate, postal consumers both large and small, will be forced to pay billions of dollars more in future rate increases that they would not have had to otherwise pay."
The Postal Service recently announced a rate increase that will go into effect if the legislation is not approved this session.
Bond has cited concerns that businesses and individuals that rely on first-class mail -- including Hallmark, which hails from Missouri -- worry the Postal Service might set prices at unfairly high rates.
A Senate aide and mailing industry lobbyists said a coalition of mailers reached a deal this fall on compromise language, but L.L. Bean -- located in Collins' home state of Maine -- rejected it.
Collins said this month it is too simplistic to view the holdup as a dispute between L.L. Bean and Hallmark.
She said Bond's language is opposed by the Postal Service, the financial services industry, postal workers' unions and other retailers, including Target and Williams-Sonoma.
She said Hallmark's proposal would continue the current "long, litigious and expensive rate-setting process."