Lambert is also keen to snap up Hoops full-back Stephen McManus, and David Fernandez as he prepares to take up his first job in management.
Maloney, 22, has been recovering from a knee injury that sidelined him for more than nine months.
Celtic boss Martin O'Neill rewarded Maloney with a three-year contract when he suffered his injury in an Under-21 game against Partick Thistle last February.
But he has slipped down the pecking order in recent months and Livingston hope the lure of first-team action with Parkhead pal Lambert will prove decisive.
Meanwhile Hassan Kachloul is unlikely to remain at Livingston next season after helping them avoid relegation from the SPL. Kachloul said: "I have done my job and might move on. It is more than likely to be in England but will definitely be in the UK."