Allardyce has been the subject of terrace chants calling for his departure from Goodison Park and there has been consistent talk in the media that Moshiri is looking to bring in a Continental coach to lead the team going forward.
"There's a huge amount of players at the club already and one of the greatest challenges will be what players are going to leave", Allardyce told radio station TalkSport.
"There are rumours about Wayne going elsewhere overseas - in China and America - but rumours are rumours", he told TalkSport.
"I wouldn't consider anything on the cards on the moment", Allardyce said of Rooney.
Rooney's 53 goals in 119 appearances with England make him the Three Lions' all-time leading goal scorer.
Brandon man killed in Maryland shooting
However, police have had "a number of contacts with this suspect in the past on a number of different issues", Chief Manger said. Police say the suspect is known to them, but they can not confirm if the shooting was domestic related.
As was the case a year ago, Rooney has no shortage of offers to move overseas, lucrative deals possible in the United States or elsewhere.
"Do I want him to stay?"
"Wayne in himself, on a day to day basis, loves his football, he loves training, he wants to play and play well like everybody else".
Allardyce has started Rooney in 16 of his 23 Premier League matches in charge of Everton.
Indeed, as it stands Bookmakers believe that Rooney is most likely to remain on Merseyside, with PaddyPower listing him at 4/11 to remain whilst they offer 9/4 to play for any MLS club.
Juventus boss Massimiliano Allegri says he will have talks with the Italian club at the end of the season amid speculation that he will take over as Arsenal manager.