Coote, 42, was sacked by the PGMOL, the referees' body, in December after appearing in several leaked clips, including one in which he insulted former Liverpool manager Jurgen Klopp.
David Wagnee, Ian Evatt, Adin Osmanbasic, Des Buckingham and Jim Goodwin have all been touted as possible candidates, but Wilshere is seen as the odds-on favourite for the role.