Rooney's representatives have been in the United States to negotiate terms and the former England skipper is willing to leave the Premier League club.
Rooney is the club's top scorer this season with 11 goals but, as an indicator of where he is and where the club's form is, he has not scored since December 18.
With Manchester United, Rooney was 2010 Footballer of the Year and won five Premier League titles.
It is understood Rooney has been offered a contract until the end of the 2020 MLS season.
The MLS trade and transfer window re-opens on July 10 and runs until August 8.
Earlier this week, Allardyce said he wanted England's and Manchester United's record goalscorer to see out the final year of his contract next season and added: 'There have been rumours about Wayne going overseas, China or America.
Wayne Rooney's rumoured MLS move could be the right way for the former England captain to prepare for retirement, according to Gary Neville.
However, further reports have suggested Rooney would prefer to stay at Goodison Park this summer and could yet shelve his plans to leave.
He reacted angrily to being substituted against Liverpool in a 0-0 draw last month.
Kenwright confirmed at Everton's end of season awards that Rooney had taken a sizeable pay cut to return to his boyhood club last summer, and described negotiations for that deal as his "greatest and shortest".
Koeman was sacked in October after a nightmare start but when Rooney's final campaign in the Premier League is remembered, he was still involved in Everton's best moments this season.
West Ham manager David Moyes was in charge of Everton when Rooney was handed his first-team debut at 16.
There was also the well-documented drink-driving arrest earlier in the season, for which Rooney is serving a two-year driving ban.
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