The former England man has been reportedly homesick and wants a move to London which is closer to his home town of Ipswich.
West Ham had a bid £4million turned down from Newcastle already but they have come back with an improved offer which has now been accepted.
"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," Newcastle boss Allardyce said following a friendly defeat to Hull.
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"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."
Tottenham Hotspur were also believed to be interested in the 28-year-old but, unlike free-spending Eggert Magnusson - are not willing to match the £80,000-a-week wages he currently picks up at St James's Park.
Ruud Gullit paid £6m to bring Dyer to Tyneside in 1999, but some encouraging spells have been overshadowed by a succession of injuries and disciplinary breaches.
Dyer's repeated hamstring and groin problems reputedly resulted from poor posture, something which doctors claim to have remedied.
West Ham are looking to add depth to their midfield before the start of the season after selling Yossi Benayoun to Liverpool.
Meanwhile, Allardyce confirmed that he will look to replace Dyer quickly.
"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," he said.
"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."
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