Having snubbed Norwich City to move to Celtic on loan for two seasons, the highly rated keeper has seen his Carrow Road replacement make his way into the England set-up.
John Ruddy has been named as one of three goalkeepers - along with Manchester City's Joe Hart and West Ham's Robert Green - in Roy Hodgson's 23-man squad for the tournament in Poland and Ukraine.
The 25-year-old will link up with the England squad for their preparation matches away against Norway on May 26 and Belgium on June 2 at Wembley.
But the Football Association have given Ruddy special dispensation to miss the Belgium game so he can marry his partner, Laura.
Lennon is also keen to retain the services of Forster, but the Toon giant has spent two years knocking back Celtic's offers - even though the two clubs have agreed a £2m deal.
Lennon: "His agent is in tomorrow [Wednesday], so we are hoping we can move things on pretty quick.
"We have been delighted with him in the two years he has been here and are hoping we can bed him down for another two or three after that."
England squad for Euro 2012:
Goalkeepers - Green (West Ham), Hart (Man City) Ruddy (Norwich).
Defenders - Baines (Everton), Cahill (Chelsea), Cole (Chelsea), Johnson (Liverpool), Jones (Man Utd), Lescott (Man City), Terry (Chelsea).
Midfielders - Barry (Man City), Downing (Liverpool), Gerrard (Liverpool), Lampard (Chelsea), Milner (Man City), Parker (Tottenham), Oxlade-Chamberlain (Arsenal), Walcott (Arsenal), Young (Man Utd).
Forwards - Carroll (Liverpool), Rooney (Man Utd), Welbeck (Man Utd), Defoe (Tottenham).