The Toffees recorded a 2-1 victory over Mansfield Town in the round of 32, while Lincoln progressed to this stage by beating Morecambe on penalties following a 1-1 draw.
Match preview
Lincoln will enter Tuesday's contest off the back of a goalless draw with Wycombe Wanderers in League One on Saturday afternoon, which has left them in 14th position in the table on 27 points.
The Imps finished top of Group E in the Northern Section of the EFL Trophy, picking up six points from their three matches, to advance to the next round of the competition.
Mark Kennedy's side played out a 1-1 draw with Morecambe in the round of 32 last time out but won the resulting shootout 4-3 to progress to the round of 16.
Lincoln were knocked out of the FA Cup by Chippendam Town at the start of November, but they have been in strong form in the EFL Trophy this term and will now be eyeing a position in the quarter-finals.
Everton Under-21s, meanwhile, finished top of Group A in the Northern Section, picking up six points from theee matches, to progress to the next round of the competition.
The Toffees faced Mansfield Town in the round of 32 on November 30 and ran out 2-1 winners in spectacular fashion, with the Merseyside outfit netting twice in the final exchanges.
Indeed, Paul Tait's side were behind until the 91st minute, when Liam Higgins levelled the scores, before Tom Cannon struck the winner in the fourth minute of added time.
Everton Under-21s were last in action on December 6, recording a 7-2 victory over Colchester United Under-21s in the Premier League Cup.
Lincoln City EFL Trophy form:
W
L
W
W
Lincoln City form (all competitions):
W
D
W
W
L
D
Everton Under-21s EFL Trophy form:
L
W
W
W
Everton Under-21s form (all competitions):
W
W
L
W
L
W
Team News
Lincoln boss Kennedy could name the same side that took to the field for the first whistle against Morecambe in the last round of the competition.
Tom Hopper was only a late substitute against Wycombe in the league at the weekend, but the centre-forward is expected to be included in the first XI.
Ben House is likely to be the player to drop out in attack, with Charles Vernam and Jack Diamond set to retain their spots for the home side.
As for Everton Under-21s, Katia Kouyate was only a late substitute in his side's victory over Mansfield in the previous round of the EFL Trophy, but the attacker could come in for a start here.
Indeed, the 19-year-old was in brilliant form against Colchester Under-21s last time out, scoring three times during a strong performance.
There are not expected to be any major surprises in the away side's XI for this match, with Cannon, who scored twice against Colchester Under-21s, also set to feature in the final third.
Lincoln City possible starting lineup:
Wright; Poole, Eyoma, O'Connor, Roughan; Virtue-Thick, Sorensen, Oakley-Boothe; Vernam, Hopper, Diamond
Everton Under-21s possible starting lineup:
Zeban; Campbell, Welch, Anderson; Dixon, Price, Quirk, Hunt; Kouyate, Mills, Cannon
We say: Lincoln City 1-2 Everton Under-21s
This is a difficult match to call, and it would certainly not be a shock to see a Lincoln win on Tuesday, but we just have a feeling that Everton Under-21s will navigate their way to a narrow success in this match.