The Jewish Chronicle newspaper published the picture, which it said was taken in Dubai on an unconfirmed date.
Lingard said on Twitter: “I’ve been made aware of a photo circulating currently, that can easily be misconstrued. I want to make it clear that I do not condone any form of racism whatsoever!”
Meanwhile Rashford said: “I wasn’t aware of comments that had been made when I got pulled into a photo opportunity. Comments that I absolutely condemn. I apologise for any upset caused. We just all happened to be in the same place at the same time.
“I’d like to reinforce that I do not and will not condone discriminative language or behaviour of any kind aimed at the Jewish community or any other community.
“I truly believe that tackling anti-Semitism in and outside of the game requires a greater level of attention and should very much form part of the game’s anti-racism stance,” he added in a series of tweets.
Wiley had apologised for his actions two years ago and said he was not a racist.