However, Manuel Pellegrini and Unai Emery are also under consideration for the role.
And the club have moved quickly to offer supporters an update on how the search is going.
"West Ham United would like to assure supporters that the Board are extremely confident of appointing a high-calibre first-team manager, who has a proven record both in the Premier League and European football," it reads.
Rafael Benitez is the favourite to take over at West Ham
We have followed a process that is progressing entirely to plan and are now very close to reaching an agreement
"The Club expect to make an announcement within the next seven days, following a thorough and strategic process that has identified the best possible candidate."
Joint Chairman David Sullivan said: "I would like reassure our supporters that we have left no stone unturned in selecting the individual we believe is the right man to take West Ham forward.
"We have followed a process that is progressing entirely to plan and are now very close to reaching an agreement.
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Manuel Pellegrini is also under consideration
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"This is an exciting time for our football club. We are preparing to appoint a manager who has a proven record of success at the highest level of the game."
West Ham are hoping for a quick appointment, which may rule out Benitez, who is still negotiating with Newcastle United representatives.
Other names touted as replacements for Moyes include Slavisa Jokanovic and Paulo Fonseca.