It comes just days after West Ham chief executive Karren Brady denied her club was talking to in-demand Borussia Dortmund manager Jurgen Klopp about becoming the East Londoners' 15th full-time manager.
Gus Poyet is another name linked with the job, where he would be managing his son and current Hammers midfielder Diego.
But co-owners David Sullivan and David Gold are believed to be reluctant to part company with Big Sam unless they have a concrete agreement from one of their top targets, with Moyes said to be top of that list.
The Telegraph's report says that Tony Henry, Moyes' former Everton scout who is now in charge of West Ham's transfer targets and dealings, could help sway the Scot who has propelled his Real Sociedad side from the lower reaches of La Liga into midtable security since taking over in November last year.
The Hammers are said to have drawn up an impressive list of targets to help lure the former Preston North End boss including Real Madrid’s on-loan Javier Hernandez, Sion’s Moussa Konate, Sampdoria's Pedro Obiang and Basel defender Fabian Schär.
Moyes, who was sacked by the Red Devils a year ago this week, may be the proven manager West Ham want to guide them into the Olympic Stadium. But some fans would question whether his footballing philosophy is that different to the man they want out.
The Scot is a well-known exponent of the aerial prowess of the likes to Marouane Fellaini, a style of football which has often driven a wedge between West Ham fans and Allardyce.
But in his first press conference at Sociedad, Moyes dismissed the notion he played the percentages, insisting: "I want to play flexible, attacking football, but more than anything I want to win."