Chelsea are now leading the Premier League table by eight points and are being heavily backed to lift the league title at the end of the season due to their consistency of performance.
"I don't know about this situation", Conte said when pushed on whether a new Costa deal was imminent.
And he believes that at a club of Chelsea's stature, it is important to have that winning mentality for any competition.
One young player who could be on his way out of Chelsea is striker Dominic Solanke, whose contract is up in the summer. "I wish for him and his family the best".
Chelsea last won the league and FA Cup double in the 2009/10 season, under the tutelage of another Italian manager in Carlo Ancelotti. However, despite the clean bill of health for his squad, and a clear message that the team must not be complacent, Conte said he would take the opportunity to rest some of his key players.
"For sure it won't be easy". In the FA Cup there are 16 teams who are very strong. Our target now is to go into the next round.
"Tomorrow won't be easy".
Helder Costa scored in the third round against Stoke, then provided assists for both goals vs Liverpool in Round 4.
"We must pay great attention, because this team showed in the recent past that they have the possibility to beat great teams". In the future I hope to have many players from the academy.
"John Terry is fit for this game", Conte told a news conference on Friday.
David Luiz will be rested after playing while managing a knee injury sustained in December in a tackle which saw Sergio Aguero sent off for Manchester City.
"All the players are in good form and I have only the problem to make the best choices for the game", Conte added.