The hosts twice came close through Raul Uche Rubio, but Man Utd goalkeeper Dean Henderson frustrated the Spaniard on both occasions.
Leicester extended their unbeaten league run to four matches, but having drawn three in a row, will aim to rediscover winning ways in the first of their final three matches, against Liverpool at Holmes Park on 17 April.
Steve Beaglehole's side lie one point above the two relegation places, while United are three points better off, but have only two fixtures remaining.
United host Tottenham Hotspur at Old Trafford on 24 April before concluding their campaign against Arsenal at Emirates Stadium on 8 May.
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