Sanchez suffered a knock to his knee in Tuesday's 2-1 victory over Leicester City before being withdrawn after half-time, while Ramsey tweaked his troublesome hamstring just nine minutes after coming off the bench.
With Wilshere's return to training following a long-term ankle injury offering a boost, Arsene Wenger revealed on Friday that Sanchez was doubtful while Ramsey would miss out.
Should Wilshere feature at the weekend, it would be his first outing since he was spotted holding a shisha pipe in a photograph that emerged recently.
On Ramsey's injury, Wenger explained: "We haven't found an underlying reason [for the injury]. There's certainly a medical or biomechanical reason.
"[Wilshere] is motivated and focused to come back as quick as possible. He needs competition. Gabriel is very close to making his debut. He has the ability, good pace and he's good in the air.
"Some players will play who have not played recently but we will treat this game exactly the same as a Premier League game.
"[Wojciech] Szczesny will play. He is completely ready and determined to play."
Wenger's men will need no warning of the threat posed by Aitor Karanka's side - with the Championship high-fliers having disposed of Manchester City in the previous round.
"We won't be surprised by them. We don't have the excuse to be surprised as they beat Manchester City," the Frenchman added.
"Middlesbrough are not afraid to take on the big boys. They are well organised, play good football and they are very creative."
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