The midfielder was substituted after just nine minutes in the 3-0 win over Arsenal with a cut that left him needing five stitches.
Osman trained with the Toffees during the week and Martinez revealed that he will undergo a late fitness test ahead of the trip to the Stadium of Light.
“Leon Osman is the only doubt that we have from the squad on Sunday [against Arsenal]. The cut is healing well but the eye is not fully open,” said Martinez.
“We need to make sure that it doesn’t affect his vision, so we will have a clear idea after training today [Thursday].”
Meanwhile, Martinez also confirmed that defender Phil Jagielka will again miss out as he continues his recovery from a hamstring injury.
“With Phil being a centre-half, I want to make sure that he is 100 per cent to play 90 minutes,” added Martinez.
“We are going to organise a behind-closed-doors friendly and make sure he plays 90 minutes before he gets involved with the squad.
“He’s been reacting well to treatment and it has been great to have these couple of weeks regenerating and to have given him every opportunity to make sure he is fully fit. We don’t want one per cent of risk of re-injuring.
“I’m hoping that Phil will be available over the next 10 days but it’s dependant on the behind-closed-doors exercise.”
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