Staff at the Royal Cornwall Hospital have said they fear administrative work is going to be sent abroad.
The trust said it hoped most job losses would be voluntary
It could mean some secretarial jobs such as typing could be done in India.
Last month the trust announced 300 job cuts because of an £8.1m budget shortfall. Some wards and operating theatres may also be closed.
The Royal Cornwall Hospitals Trust said it was unable to either confirm or deny the reports and would not be drawn into a public debate.
In a statement, the trust said it was in the process of reviewing every aspect of its services.
The statement said: "As we made clear when we announced our financial recovery plan, our priority was to fully and openly involve our staff in working through all of our savings plans.
"We are focusing our efforts on doing just that, which unfortunately is leading to some proposals being shared more widely before firm decisions have been taken."
Another English health trust which tried outsourcing said it had been successful.
Peter Gibson from the East and North Hertfordshire Trust said he thought people overreacted when they first heard about it.
He told BBC News: "It's something of a knee-jerk reaction which nearly at times has racist undertones.
"What we're doing is giving personal information about patients out to third parties and the fact that it might be on the Indian sub-continent or in South Africa somehow makes it worse to some."