After Newcastle turned down a £5m bid from the Hammers on Tuesday, plus a player-exchange deal from Tottenham, the Magpies have agreed an undisclosed fee for the England international from West Ham.
Allardyce: "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.
"From our point of view and from his point of view it's a satisfactory result.
"There was a need to do it as quickly as we possibly could and we've worked all out to do that and we've finally resolved the situation.
"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."