Maximus has told the BBC it will do one million assessments this year.
Maximus is being paid £595m over three years to carry out work capability assessments for people applying for employment and support allowance.
The Department of Work and Pensions cut short a contract with Atos last year after "significant quality failures".
Maximus is promising to clear a backlog of around 600,000 claims.
It is also planning to reduce the time people wait for their results from at least 120 days to the recommended 90 days.
President of health services, Leslie Wolfe, told BBC Radio 4's You and Yours programme: "Part of what Atos didn't have was a [big] enough team to keep up with the wait times.
"That's one of our first priorities. We need to clear about one million [work capability assessment] claims this year.