Allardyce's contract is up at the end of the campaign and it is said to be 'unlikely' that he will be offered a new deal at the Boleyn Ground.
The reports states that the Hammers are already looking for a potential replacement and Moyes is at the top of the list.
The Scottish boss has only just taken up a role at Real Sociedad, but West Ham feel they could tempt him back to England should they offer him the job.
Impact on West Ham?
If West Ham do decide to go through with the move it would be a rather strange action for the Hammers to take.
One of the main gripes with Allardyce's management has been his penchant for long-ball football, but that style is exactly the same as the one Moyes tends to operate with.
The former Manchester United manager has proven to be excellent at forming organised and efficient teams in a rigid tactical structure, but it is not a particularly aesthetically pleasing style.
It means that his appointment would be one which wouldn't progress the side enough to galvanise the fan-base and unite them in support for their boss.
West Ham have traditionally played expansive, attacking football and it will be important that whoever their new manager is ensures that their tactics are fitting with these principles.
Moyes is unlikely to be able to do that sufficiently and if West Ham are intent on changing their manager at the end of the campaign, then they should look elsewhere.