Enrique is set to join from Villarreal for a £4.2million fee, and Allardyce has indicated that they could soon go back in for Serbian defender Ivica Dragutinovic.
"When we get Enrique signed it will finalise an excellent few days for us," Allardyce told the Evening Chronicle.
"As far as Dragutinovic is concerned, if it is still a possibility then I will try and pursue it this week.
"In Enrique we have signed a player who has huge potential at a very good age and who has already broken into the first team at Villarreal."
But signing Enrique is seen by Allardyce as the real coup: "Either our chief operating officer Russell Cushing or chairman Chris Mort got a phone call which said the player was available.
"I asked the price and said 'go' and they went. And in 24 hours he was here.
"Russell Cushing and Chris Mort are the reasons Enrique is signing for us rather than Liverpool or Manchester City."