The Premier League player, who can’t legally be named, was held by police last Friday before being released on bail, WonderTV Media learnt.
Cops also raided his home earlier this month where “several items were seized”.
The International player has now been suspended by Everton while police investigate the allegations.
A statement from the team said: “Everton can confirm it has suspended a player pending a police investigation.
“The Club will continue to support the authorities with their inquiries and will not be making any further statement at this time.”
His arrest has sent shockwaves through the club with the Toffees’ first game of the season against Southampton just weeks away.
A source said: “It has stunned the dressing room. Everyone’s hoping the allegations aren’t true.”
The alleged offences have been described as “very serious” with the probe led by Greater Manchester Police’s Major Incident Team.
In a statement, the force said: “Officers arrested a man on Friday 16 July 2021 on suspicion of child sex offences.
“He is on police bail pending further enquiries.”
Everton are due to fly to the US to compete in the Florida Cup against Millonaros on July 26.
The match will be the first with new manager Rafa Benitez at the helm after he replaced Carlo Ancelotti.
Benitez’ appointment sparked outrage due to his history with arch rivals Liverpool.
Speaking on his appointment, the 61-year-old said: “I am delighted to be joining Everton.
“Throughout this process I have been greatly impressed by the ambition shown by the senior representatives at the Club and their desire to bring success to this historic Club.
“I believe this is a Club that is going places. I’m determined to play a big part in helping this great Club achieve its ambitions.”