West Ham co-owner David Sullivan remains on good terms with Benitez and said on Thursday that the club were "very close" to appointing a "high calibre" manager to succeed David Moyes at the London Stadium.
Express Sport understands Benitez is open to managing the Hammers if they give him a £100m transfer budget, total control over signings and a £6m-a-year contract.
Having announced on Wednesday that Moyes would be leaving and that they planned to have a new manager in place within 10 days, they have little over a week to find their man.
Who will replace David Moyes at West Ham?
"They can't make assurances to him again and not back it up because the biggest danger for the club is him walking away".
Newcastle manager Rafa Benitez has done a brilliant job with the Magpies over the past few years, but his excellent work has not gone unnoticed.
Joint Chairman David Sullivan added: "I would like to reassure our supporters that we have left no stone unturned in selecting the individual we believe is the right man to take West Ham United forward".
Their preference is to hire a manager who has worked in the Premier League, although they have not denied making contact with Shakhtar Donetsk's Paulo Fonseca last week. Where do you want to be? Will it be a mid-table or...
Waddle admitted Newcastle need to splash the cash in order to compete in the Premier League though.
Although he is resisting the temptation to leave, Benitez would be intrigued by the West Ham job if he can not reach a satisfactory agreement with Newcastle.
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