Shakespeare took charge of the Premier League champions following the decision to part company with title-winning manager Claudio Ranieri.
Having won his first match at the helm against Liverpool at the end of last month, Shakespeare has added two wins from two Premier League games in March - and has also guided the Foxes into the Champions League quarter-final.
Jurgen Klopp is also nominated, Liverpool having bounced back from a poor spell to secure March home wins over Arsenal and Burnley, plus a good point at Manchester City last weekend, and they are now fourth in the Premier League table.
Bournemouth also managed two wins and a draw in March, pulling nine points clear of the drop, and manager Eddie Howe is also among the nominations.
The other nominees are Sam Allardyce (Crystal Palace), Antonio Conte (Chelsea) and Mauricio Pochettino (Tottenham), who all led their clubs to two wins from two Premier League games in March.
The winner will be announced at 1100 GMT next Friday.
The Player of the Month nominations are Bournemouth duo Artur Boruc and Josh King, in-form Everton forward Romelu Lukaku, Southampton's Nathan Redmond, Crystal Palace defender Mamadou Sakho and Liverpool's Georginio Wijnaldum.
King and Lukaku both banged in four March goals each, while winger Redmond and midfielder Wijnaldum bagged two apiece.
Boruc made 12 saves and kept a clean sheet during Bournemouth's unbeaten month while Sakho and the Palace defence combined to shut out West Brom and Watford, as the Eagles secured a pair of crucial wins.
The Goal of the Month nominees are Fernando Llorente (Swansea v Burnley), Andros Townsend (Crystal Palace v West Brom), Oumar Niasse (Hull v Swansea - first goal), Harry Kane (Tottenham v Everton), Eden Hazard (Chelsea v West Ham), Robert Huth (Leicester v West Ham) and Jesse Lingard (Manchester United v Middlesbrough).
