Missing out on Vargas represented more transfer frustration for Poyet, who is still waiting to hear whether Fabio Borini will be returning to the Black Cats after a £14m fee was agreed with Liverpool.
Poyet has revealed, however, that there is no deadline as yet on the Italian to make his decision.
He told reporters: "We were very close to getting Vargas. It was a matter of decisions between everybody - clubs and cities, living and possibilities and things, and in the end, everything didn't come through for us to get the player.
"That was close to us making a decision on whether Fabio was coming or not coming. It's daily. We haven't put a date on it, but it's daily and we are looking for options."
Vargas's parent club Napoli are also believed to be in the race to sign Borini, who could be surplus to requirements at Liverpool once the prospective transfer of Mario Balotelli from AC Milan is confirmed.