Nuno Espirito Santo and David Moyes have emerged as two of the leading candidates to replace Carlo Ancelotti as Everton manager.
However, he has quickly turned his mind to who might replace the Italian.
At a meeting of senior figures at the Goodison Park club, it is understood the names of former Wolves manager Nuno and West Ham boss Moyes were prominent.
Moshiri was impressed by the job Nuno did during his four seasons at Wolves, while Moyes was close to being re-appointed as Everton boss when Ancelotti became available in December 2019.
It is understood Moyes has agreed a three-year contract to remain at West Ham, but that it is yet to be signed.
It is not known whether the Scot would be willing to turn his back on the Hammers if a chance of returning to Everton – the club where he had so much success before leaving in 2013 to replace Sir Alex Ferguson at Manchester United – was offered.
Nuno is known to be keen on an instant return to the Premier League after his exit from Wolves and has had discussions with Crystal Palace about their vacancy.
Ancelotti left Everton to succeed Zinedine Zidane and become head coach of Real Madrid for a second time, after 18 months in charge at Goodison Park.