The case of Catholics Mary Doogan and Connie Wood against Greater Glasgow Health Board will be heard later.
The pair had previously won a court challenge after objecting to the health board's request that they 'delegate, supervise and support' midwives in carrying out abortions.
But Greater Glasgow Health Board is now appealing that decision before Lady Hale, Lord Wilson, Lord Reed, Lord Hughes and Lord Hodge.
A final decision is expected to be made today.
After ruling in their favour in April 2013 Lady Dorrian said: "In our view the right of conscientious objection extends not only to the actual medical or surgical termination but to the whole process of treatment given for that purpose."
She added: "The right is given because it is recognised that the process of abortion is felt by many people to be morally repugnant."
Premier has asked Greater Glasgow Health Board to comment on today's hearing but has not received a reply.