Having already landed Fiorentina full-back Marcos Alonso, Poyet is now hopeful the Barclays Premier League strugglers will seal the deal for Argentinian Vergini over the weekend.
Poyet said: "Normally I don't comment but there have been so many comments from the other side of the Atlantic that they have put me in the position where I have to say we are interested.
"I thought we were very close yesterday but not close enough.
"He is a central defender who has been playing at the top level for a few years and he has had a chance to play with the Argentina national team. He's big, so he's good for England, he reads the game well and he is a good defender. Let's hope he can adapt quickly if we get him."
Poyet refused to comment on other specific transfer rumours but did confirm out-of-favour midfielder Cabral has joined Italian side Genoa on loan until the end of the season.
The Cape Verde international was signed by Poyet's predecessor Paolo Di Canio on a free transfer from Basel in the summer and impressed on his debut against Fulham, but quickly fell out of favour.
Poyet added: "That is complete and Cabral is going to be in Genoa from today.
"It was difficult for him to play - sometimes there is not a clear reason but there are other players in his position and we were looking for special things and he didn't work, for me at least.
"Now he's got a very good opportunity to play in Italy with a very good club and I hope everything goes well for him."