The clubs have agreed an undisclosed fee for the England international, who wants to be closer to his family, and Allardyce admits a deal is in the offing.
"We've agreed a deal so you'd have thought that he will finish up where he needs to finish up to solve his family problems," Allardyce said following a friendly defeat to Hull.
"From our point of view and from his point of view it's a satisfactory result.
Although he has not officially left St James' Park, Allardyce has already turned his attention to replacing the versatile 28-year-old.
He said: "My job now is to replace him, which shouldn't take too long I don't think, and then move on to other players in other positions as quickly as I can or at the same time.
"I think we'll be close by the end of the week to maybe one or two players, I would hope.
"Then after that I would look to get at least another two or maybe three in because if Kieron goes, whoever comes in to replace him, that's only one for one, so it's not actually improving the squad in terms of strength in depth.
"There's still three or four players need to be got on top of whoever we replace Kieron with."