The 21-year-old played an integral role in keeping the club in the Premier League last season, scoring five goals in three games in April as Sunderland pulled off a great escape.
|11/4||Sunderland are 11/4 with Paddy Power to finish in the top 10|
And Poyet has warned the former Ipswich Town man that he will be sold if no agreement is reached.
The Uruguayan told the Sunderland Echo: "For me, Connor's situation is clear, he needs to sign a new contract or we let him go - we sell him.
"There is nothing in between. He is an asset to the football club and we paid a lot of money for him.
"Last year, he was a key part of us staying up and he was very important but he needs to realise that the club needs to look after the interests of the football club as well. I suppose that, one way or the other, we are going to have an agreement.
"Which one of those [two outcomes] is going to happen, I don't know -- it is not my job. That is (sporting director) Lee Congerton's department."