The Scotsman was appointed on a short-term deal in November, which expired at midnight following their final day win over Everton at London Stadium.
Moyes has been in contact with the board since Monday over his future, while the board are also meeting other candidates, with Shakhtar Donetsk manager Paulo Fonseca having met co-owner David Sullivan at his Essex home this week.
However, while the West Ham hierarchy are speaking to other candidates, it is understood that Moyes is still within the frame for a new deal, and Gold hopes he stays put.
"I've enjoyed David Moyes, I've enjoyed his company, I've enjoyed working with him, he's a proper professional - his record speaks for himself," said Gold.
"As we speak David Moyes is on the list of managers and we'll be selecting one in due course. I'd like to think it was David Moyes, I really hope so."
It was a troubled season both on and off the pitch for West Ham. However, in terms of their footballing problems, Gold puts it down to one key factor.
"We were disappointed at our performance right the way through the year. We changed our manager which is always a difficult time for us," Gold added.
"When I look back, I'm looking to see why did it happen? Was it because of the manager situation, was it because the players weren't good enough?
"In actual fact, what I have discovered, if you look at the injury list right the way through the season, there was a time just before Christmas we had 12 senior players injured, unavailable for selection.
"No team can suffer that kind of situation."
The Hammers board hope to have their new manager appointed in quick fashion, but are expecting negotiations to go into next week.