The Old Trafford stalemate means that Liverpool are one point clear of second-placed Manchester City with 11 games of the season left but they have drawn three of their last four matches.
Ole Gunnar Solskjaer’s team created the first opening during a high-tempo start to the encounter but Marcus Rashford’s shot was gathered by Alisson.
When Liverpool got a foothold in the match Mo Salah failed to hit the target with a free-kick from just outside the area while United suffered a blow midway through the first half when Ander Herrera had to limp off the pitch and be replaced by Andreas Pereira.
Solskjaer had to make another change with only 25 minutes on the clock when Juan Mata was replaced by Jesse Lingard while Liverpool also had to make an enforced substitution before the interval with Daniel Sturridge coming on for Roberto Firmino.
Liverpool edged the first period but failed to trouble David de Gea in the United goal while the home team almost took the lead just before the break only for Alisson to deny Lingard after good build-up play by Paul Pogba and Romelu Lukaku.
Lingard was injured in the process and had to be replaced by Alexis Sanchez as United made all three of their substitutions before half time.
Pogba forced Alisson to make a smart save soon after the restart as United began the second half well although the Frenchman was adjudged to be offside, while Rashford poked the ball wide as Solskjaer’s team began to get on top.
United also defended stoically and Jurgen Klopp’s team grew increasingly frustrated as the match wore on with Georginio Wijnaldum lashing a 65th-minute effort high and wide as the teams cancelled each other out.
United thought they had grabbed a winner when Chris Smalling’s cross was turned into his own net by Joel Matip but the effort was ruled out for offside.
A draw was probably a fair result as both sides struggled to create many openings although United almost snatched a last-gasp winner when Smalling narrowly failed to get on the end of Lukaku’s cross.