There have been strong reports that a £12m deal between the parties has been agreed in principle, in a move that would see the Everton forward move to Major League side side DC United in the next transfer window.
Rooney's agent Paul Stretford had been in talks with several MLS clubs before he joined Everton last summer.
Allardyce's future is also uncertain at the club, with fans not happy with his style of play.
According toThe Sun, per the Mirror, Rooney is somehow still earning £15 million-per-year (£300,000-per-week) at Everton and he would demand the same kind of salary in the MLS.
It is understood the deal is for two-years and will cost DC United £12million as Rooney has chose to bring down the curtain on his second Everton career, having return to his boyhood club last summer following 14-years at Manchester United.
"It would have to be massive for him to want leave Everton".
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The Croxteth-born product of Everton's youth system is the Blues' leading scorer this season but has struggled for form since the turn of the year and hasn't found the net in 2018 since dropping into a deeper midfield role.
The rumours of Rooney, at some point, heading for a big payday in the MLS or China have been around for a couple of years but only in recent months have they returned.
In a nutshell, if Rooney completes a move to DC this summer, three separate clubs will be paying his yearly salary.
However the 63-year-old signed a contract that runs till June 2019 and is privately mentioned to have grown assured of remaining in cost because of the facet's newest outcomes.
'So I would not take into account something on the playing cards on the second.
Chairman Bill Kenwright, who was instrumental in brokering his return to his boyhood club, is keen for him to stay while owner Farhad Moshiri is thought to be ready to let him go if the player wants out.