Manchester United are being linked with Everton pair Stephen Pienaar and Jack Rodwell.
The Telegraph says United are believed to be considering moves for both teenage midfielder Rodwell - also a target for Arsenal and Chelsea - and South Africa international winger Pienaar.
Rodwell, who signed a five-year, £25,000-a-week deal last season, will be offered an improved contract to stave off United's interest, while Moyes remains confident of securing Pienaar's future before the start of the World Cup in the player's home nation.