But supporters of the Senate legislation said they hoped to keep their central principles intact.
The Senate legislation would have experts decide later how the machines should accommodate the disabled.
It takes 60 votes to cut off a filibuster and move forward with Senate legislation.
"It is a positive response to many of the concerns we had with the Senate legislation," he said.
Senate legislation is expected to be introduced next week.
He drew up Senate legislation to prevent the Administration from doing that.
The temporary worker program in the Senate legislation, however, needs improvement.
Democrats today introduced their own bill, which is similar to the Senate legislation.
Some lawmakers believe that the Senate legislation still does not go far enough.
The initial Senate legislation had been passed last July by a vote of 71 to 27.