Allardyce is already said to have told Moshiri that Rooney does not figure in his plans next season, but there are reportedly figures in the Everton boardroom who want to keep the fan favourite.
He also won five Premier League titles and the Champions League.
The 32-year-old returned to the club he left in summer 2004 intent on driving Everton forward into a new era of success under then manager Ronald Koeman.
The former Manchester United striker's return to Goodison Park started brightly enough but has fizzled out, and he recently made headlines for his angry reaction to being subbed by Allardyce during the goalless draw with Liverpool.
"Then he goes to Manchester United and has an incredible career".More news: Ex-Manchester United boss Alex Ferguson showing signs of recovery
It is understood Rooney has been offered a contract until the end of the 2020 MLS season. Then he's in the Premier League again with Everton, so I think his career in England was ideal.
"Now he goes to MLS and there have been some iconic players in that league". You have Gerrard, Lampard, Vieira, so many of them.
"So I just hope he enjoys and of course he will bring the attention that MLS needs to attract more people to steal people from the sports the Americans normally are in love with".
Reports emerged Stateside earlier this week suggesting that Rooney had opened discussions with the Major League Soccer side over a move later this month.
West Ham manager David Moyes was in charge of Everton when Rooney was handed his first-team debut at 16.