The 20-year-old, who is out of contract this summer, had become a Hatters regular this season before recently being sidelined by a a cracked metatarsal.
Lacey scored once in 19 games before being forced off against Southport on November 26, but he is set to resume training this week.
The online report read: “Sky Sports understands a host of Football League clubs are monitoring the situation of young Luton centre-back Alex Lacey.
“Bournemouth are long-term admirers, while fellow Championship sides Charlton and Millwall and League One Peterborough have also sent scouts to watch the talented 20-year-old.
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“With his contract set to expire in the summer, the Hatters are keen to try and get him to pen a new deal to prevent losing the youngster for a knock-down fee.”