But Greenwood emerging on the right wing and the COVID-19 pandemic may slow down their thinking.
Recent reports suggest United want to pay half of what Dortmund want for Sancho. And O'Shea thinks Greenwood may factor into that thinking.
The Irishman told talkSPORT of Greenwood: "He has a long way to go before he's getting up to the [standard of Wayne] Rooney, or [Ruud] van Nistelrooy or Solskjaer. But he is on the right track.
"Look, if he can maintain that and make sure at the moment Ole is picking him in the team, he might be changing their mind about the transfer targets as well.
"If you bring in a striker, do you upset the balance at the moment with the three of [Anthony] Martial, [Marcus] Rashford and Greenwood?"