Senate Republican Leader Bill Frist it would be another week before the version of the bill approved by the Senate Judiciary Committee last Thursday, with a number of amendments, was put in final written form.
The proposal to take asbestos lawsuits out of the courts and create a fund of up to $153 billion to pay victims narrowly passed the Republican-run committee after three final days of lengthy and sometimes rancorous debate.
But neither Democrats nor Republicans are entirely happy with the measure and both want to make changes to the bill when it comes to the floor.
Insurers think they are bearing too big a burden and organized labor says the payouts are not generous enough. Both groups oppose the current measure, which may not have the 60 votes needed to overcome procedural hurdles in the Senate.
The Senate's last scheduled day before the summer break is Aug. 1 and the chamber returns to work on Sept. 2.
Asbestos was widely used for fireproofing and insulation until the 1970s, when scientists concluded that inhaled fibers could be linked to cancer and other diseases. Lawsuits over the mineral have driven 67 companies into bankruptcy.