The veteran Czech Republic international has been on the radar of the Everton boss ever since he returned to Merseyside, and last summer was apparently trying to sign the player Moyes originally brought to England in the first place, when he was the East London club's manager.
Midfield, though, does not appear to be the number one transfer priority for the Toffees this coming summer, with Idrissa Gueye, Kiernan Dewsbury-Hall, and new England international James Garner all performing very well in the team's engine room.
In addition, last year's deadline day addition, Merlin Rohl, and impressive youngster Harrison Armstrong are also available in this position.
Full-back, another wide forward with Jack Grealish likely returning to Manchester City at the season's end, and a striker who can be relied upon to consistently turn the chances the Blues are creating into goals, would be the main focus for the next transfer window.
However, Gueye is 36, and although at the moment he appears to be ageless and able to carry on playing at an elite level for the foreseeable future, eventually time will surely catch up with the Senegal star.
Soucek is over thirty himself, so he does not represent a young alternative to Gueye, but he is a good, consistent, and very experienced defensive midfielder and one that Moyes obviously knows, likes, and trusts.
The Hammers also look ready to sell him in the summer, whether they survive relegation or not, and cash in on a player who is one of their more attractive assets.
Ideally, I think the Toffees should be looking at younger alternatives if they are to bring in another midfield player, or perhaps trust the current group, particularly Rohl, who was presumably signed to become a long-term replacement for Gueye.
This is still a transfer that seems less likely to happen than others, given that Everton will have to focus their resources on the other pressing needs to strengthen the squad in preparation for next season, and possibly, if they qualify, a challenge in Europe.