"We talked about a position the other day and considered four players," Poyet was quoted as saying by British media.
"I picked two. It was simple. One of the other two (selected by De Fanti) was a definite no. If that one is coming, I won't be here. The player has to be the right one for me. If not, I'm not going to accept it.
"They need to give me the options and I need to pick the ones I like. We need to see if they are available and if they fit into the financial possibilities, then we will move forward.
"I'm sure it's going to be done in the right way, in the common sense way. Everyone has a responsibility."
Italian Paulo Di Canio started the season as Sunderland boss but he was less than impressed by De Fanti's work, complaining that he had chosen none of the 14 players signed by his compatriot.
The volatile Di Canio failed to mould the new recruits into a squad he barely kept up the previous season and was dismissed in September with the side bottom of the table.
Former Sunderland midfielder Kevin Ball then took charge on a caretaker basis before last month's appointment of Poyet, who has enjoyed a promising start with home wins over northeast rivals Newcastle United and big-spending Manchester City.
However, the former Real Zaragoza, Chelsea and Tottenham Hotspur midfielder was fully aware of his squad's deficiencies, which he had told De Fanti of, and said new signings were required if they were to improve.
"I've told him my opinions about everybody here already and I've told him what I think we should be looking for," Poyet said.
"Roberto and Valentino will manage it and then I will start checking on them every now and then - I like to go and see players live, if it's possible. The players already here have an opportunity now but there are others I think we need.
"The idea is to start moving. The sooner we move, the sooner we get a player, the better for me," the former Brighton and Hove Albion manager added.
Despite the threat of leaving, Poyet said he was optimistic that he could work with the Italians.
"The sporting director exists everywhere in the world. I know in England it's not very common. The manager is normally in charge of everything but I had it at Zaragoza as a player and my situation is clear.
"So far, everything is working fine. The connection and communication have been great."
(Writing by Patrick Johnston; Editing by Peter Rutherford)
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