The Hammers are still managerless, with Avram Grant expected to be confirmed as Gianfranco Zola's successor next week.
But Sullivan, who bought the club along with David Gold last season, has already been linked with names such as Thierry Henry and confirmed his desire to bring LA Galaxy star Beckham back to his native east London on loan.
"I'd love David Beckham," Sullivan told the Daily Mail. "He has massive cult status in east London and still has a house near Bishop's Stortford, not far from our training ground. He's from Chingford, which is our heartland. We are the club of east London and Essex.
"If AC Milan didn't want him in January and if he still wanted to do it, I'd love to have him. He seems a remarkable professional."
As for Cole, who began his career with the Hammers before moving to Chelsea - where he is out of contract this summer - Sullivan said: "We've asked Joe to name a price but he said no."