Juan Mata is expected to try to leave Manchester United in the current transfer window after he was brought on and taken off in the space of 30 minutes by Jose Mourinho in the Community Shield.
The Times says Everton are exploring the possibility of signing the Spain international after completing the transfer of Swansea's Ashley Williams on a two-year contract.
United have maintained that they will not sell the 28-year-old, but his future has been thrown into doubt after he appeared unhappy at being replaced half an hour after coming on in the 2-1 win over Leicester City on 7 August.
Mourinho said he made the change because Mata was the "smallest player" on the pitch and that he needed more height in the team to defend long balls.
"The rules allow six changes, I had made five and I wanted to stop the game," he was quoted as saying by the BBC.
"I think he played very well. He gave me exactly what I needed."
Mata was sold by Mourinho during his second spell at Chelsea to United in January 2014, amid reports that he was unhappy with the Spaniard's lack of pace and work rate.
He featured in every league game under Louis van Gaal last season and scored in the FA Cup final against Crystal Palace in May.
But the arrival of Henrikh Mkhitaryan from Borussia Dortmund has pushed Mata down the pecking order at Old Trafford, making a summer exit from the club a likely outcome.
Everton are bound to face competition from other clubs in England and in Europe for the former Valencia midfielder, who has 41 caps for Spain.
Meanwhile, the Times says Mourinho is also considering selling Daley Blind after renewed interest from Inter Milan in the Dutch defender, following Frank de Boer taking charge of the Italian club this week.