Allardyce is aware of N'Zonzi's qualities from his time in charge of Blackburn and subsequently having watched the France international at Stoke City before his move to Spain.
The 28-year-old found success at Sevilla and was even a target for Juventus last summer, instead signing a contract extension at his current club until 2020. After falling out with coach Eduardo Berizzo, he has reportedly been forced to train alone and has been left out of the squads to face Villarreal and Cartagena.
Everton boss Allardyce will assess his options as the January window draws closer and is reportedly willing to pay up to £25 million for the midfield dynamo.