Rangers slumped to a new low on Friday night after losing 2-1 away to bottom-of-the-table Dundee for their second successive defeat under caretaker boss Graeme Murty.
Alves, who has 94 caps for Portugal, played no part in either defeat due to a back injury but his return can't come soon enough as the Gers look to stem the tide of sluggish defeats.
The 35-year-old was signed by Pedro Caixinha, who was axed last month, and Sutton told BT Sport that he thinks the veteran centre-back probably wants to quit the club when the transfer window opens.
"Alves is an interesting one at the moment for me. He's injured, but you can see him thinking 'in January, I want off'."
Sutton might be right. This isn't quite what Alves signed up for and, being in the twilight of his well-travelled career, the big man might fancy another, brighter move before hanging up the boots.
His importance to Rangers has been highlighted since falling injured and it's probably no coincidence that the Gers have lost both games that he missed through injury.
They host Aberdeen at Ibrox on Wednesday night and fans will be hoping he's back in time.