David Moyes, who returned to the Hammers on December 30 following the sacking of Manuel Pellegrini, named the former England midfielder as one of the candidates to join his coaching staff on Thursday.
Cole told BT Sport: "There's been no contact – nothing.
"It's a tough one because I enjoy sitting here on a Saturday morning.
"I enjoy my spare time. Who knows what's happening? At the moment I am happy here."
Moyes has appointed regular sidekick Alan Irvine as his assistant manager since returning to the club and is considering adding Cole, who came through West Ham's youth system before making 157 appearances for the club across two spells, to his staff.