Senate Majority Leader Bill Frist said on Tuesday he was confident a bill setting up a national fund to pay asbestos claims would be on the Senate floor soon, despite criticism by some insurance companies.

“I am confident we will have a bill, a bipartisan bill, that centers on a trust fund, on the floor of the Senate in the not too distant future,” Frist, a Tennessee Republican, told reporters in a briefing on the Senate floor.

He said he was not discouraged that some insurance companies had written to express unhappiness with the draft bill, noting the battle over the details had lasted two years.

The bill is being drafted by Sen. Arlen Specter, a Pennsylvania Republican. Some insurers that fired off the latest complaints to the Senate, such as American International Group Inc., are long-time critics of the plan.

Specter wants to take claims out of court and pay them from a $140 billion trust funded by business and insurers.

Shares of U.S. companies fighting asbestos suits fell Tuesday on news of the insurers’ complaints. Building materials company Owens Corning fell 17 percent. Shares of W.R. Grace and Co., fell 4.5 percent.

Frist said progress had been made on the asbestos proposal recently but the difficulty was getting bipartisan backing.

Specter has been struggling to get the support of Republicans who share insurers’ concerns that the plan may not adequately limit companies’ asbestos liabilities.

Democrats are worried the fund may run out of money and may not adequately compensate some victims.

Asbestos is linked to cancer and other diseases. Thousands of injury claims have forced many companies into bankruptcy.