Allardyce, who replaced Ronald Koeman at Goodison, took charge of the club when the Toffees were in a very difficult position in the league and battling against relegation.
The former England manager succeeded in guiding Everton away from relegation trouble and they look well settled to finish within the top 10 this season.
However, he is under pressure of discontent from the fans. Allardyce, who has 12 months left on the contract he signed in November, is not a popular figure among the fans, and his style of football does not appeal to supporters.
The report claims that the Porto coach is viewed as a leading contender. He has enjoyed a superb first season with Porto, leading them to their first league championship in four years.
Furthermore, the 43-year-old is likely to welcome Everton’s interest. Everton would comfortably be able to double his £1.7 million salary and pay the release clause on the year that remains in his contract.