Frank Lampard hopes Chelsea team-mate John Terry can follow him by signing a new contract with the club.
Argentina striker Sergio Aguero insists he is happy at Manchester City.
As revealed on Friday, Forest were poised to sign Gillett on a free transfer after he impressed on trial.
Gillett left Doncaster earlier this summer upon the expiration of his contract at the Keepmoat Stadium.
The 26-year-old has penned a two-year deal at the City Ground after being reunited with former Rovers boss Sean O'Driscoll.
Gillett is Forest's fifth signing of the summer and Forest chairman Omar Al Hasawi was delighted to land the midfielder.
"Simon is a player that Sean knows well from his Doncaster days and we are delighted to welcome him to the club," Al Hasawi told the club's official website.