The Old Trafford starlets are set to follow in the footsteps of previous United youngsters Daniel Drinkwater and Tom Cleverley, who moved to the Midlands club to gain first-team experience earlier in their careers.
The 19-year-old pair are highly regarded by the Manchester United hierarchy, and Sir Alex Ferguson is believed to be keen for the duo to continue their footballing education with the high-flying Foxes.
Keane has made two appearances for United this season, both in the Capital One Cup, against Newcastle and in the defeat to Chelsea last week.
Lingard, who operates as a midfielder, is yet to make his competitive debut for United, but starred during the club's worldwide pre-season tour this summer.
Leicester boss Nigel Pearson is expected to slot the talented duo into the Championship side's squad to face Bolton at the Reebok Stadium on Tuesday night.
The Foxes are currently fourth in the Championship table, four points off the lead, and Pearson is thought to view Lingard and Keane as important additions as he looks to build a team capable of winning promotion to the Premier League.
Manchester United have promoted Danny Welbeck and Cleverley to the first-team after both enjoyed successful loan spells in the lower leagues, and Ferguson is thought to be eager for Keane and Lingard to continue their development away the glare of the Old Trafford spotlight.