Wilshere has been strongly linked with a move to West Ham, the club he supported as a boy, after entering the final year of his contract at Arsenal.
England's Jack Wilshere
Speaking to Talksport in separate interviews on Friday, West Ham legends Tony Cottee and Tony Gale both advocated the England midfielder's signing - a view which appears to be shared by many Hammers supporters.
However, when asked if his side have plans to bring Wilshere across London ahead of September's transfer deadline, Sullivan, the West Ham co-owner, told the same radio station this morning (Tuesday): "I can’t comment on players who are at other clubs. But what I will say is that midfield is not an area we are looking to strengthen."
Nevertheless, the Welshman did offer a glimmer of light to those hoping to see Wilshere in Claret and Blue, adding: "You can never say never."
West Ham have enjoyed a productive window so far, bringing in established Premier League quartet Pablo Zabaleta, Joe Hart, Marko Arnautovic and Javier Hernandez.
Whether Wilshere will join them remains to be seen, however, the chances of him leaving Arsenal after 16 years appear to be growing increasingly likely, amid reported interest from several English and continental sides.
Do West Ham need more midfielders and, if so, should they target the Arsenal man?