Aside from sitting sixth, Ralf Rangnick's team has Arsenal and Tottenham ahead of them with at least a game at hand while West Ham could overtake them when they play Everton on Sunday.
Kelechi Iheanacho put the visitors in front in the second half but although Fred equalised just three minutes later, neither side could find a winner.
"In the last 15 minutes we tried to score a goal but in this game, we did not score the late goal," he told Sky Sports. "We don't have the offensive players and even Jesse Lingard was feeling sick.
"As long as it (top four) is possible mathematically, it is possible. It is our job and duty to do our best to finish on the best possible note."
United are hunting for a new permanent manager, with Ajax boss Erik ten Hag emerging as a clear favourite, and the team's shortcomings were all too obvious against Brendan Rodgers' team.
The home side, missing top-scorer Cristiano Ronaldo through illness, looked anaemic in attack for most of the first half as they struggled to establish any attacking rhythm.
Bruno Fernandes, who signed a new deal this week, was put through by Fred in the 27th minute but the Portuguese playmaker failed to make good contact and Kasper Schmeichel saved with his outstretched left foot.
Minutes later Leicester had a chance to take the lead but Iheanacho failed to control the ball on the edge of the area and ended up skewing his shot wide.
The sides remained deadlocked at 0-0 at the interval.
United looked sharper in the early stages of the second half but midfielder Scott McTominay was lucky to stay on the pitch after a nasty-looking challenge on James Maddison.
Marcus Rashford was brought on by Rangnick to replace McTominay and add extra zip to the home side's attack.
But it was the visitors who took the lead in the 63rd minute when a United attack broke down and Maddison attacked down the left.
The England international produced a fine left-footed cross and former Manchester City forward Iheanacho headed past David de Gea, who got a hand to the ball but could not keep it out.
Leicester's lead was short-lived, however, with United level just three minutes later.
The ball broke for Fernandes after the visitors failed to clear their lines properly. Schmeichel could only parry the shot from the Portuguese forward and Fred lashed into the roof of the net.
Leicester had chances to re-take the lead.
Inheancho lifted the ball over the crossbar from a tight angle and De Gea then produced an outstanding fingertip save to keep out a goal-bound header from Wesley Fofana.
Leicester continued to look the more dangerous side and had the ball in the net with 10 minutes to go when Maddison finished at the back post.
But the goal was overturned for a foul by Iheanacho on Raphael Varane.
United piled on the pressure late on but could not find a winner.