Tigers match captain Slater has been shortlisted along with Sale Sharks duo Danny and Dan Braid and Worcester Warriors full-back Chris Pennell. The winner will be voted for by a panel of media professionals and the result announced on Wednesday.
Tigers winger Goneva was named the Aviva Premiership Player of the Month for February and received £250 from title sponsor Aviva, which he donated to the Matt Hampson Foundation.
Goneva and Slater, along with Logovi'i Mulipola, are in the running for the Tigers' Goldsmiths Player of the Month award. The trio have been shortlisted after their excellent performances in Tigers' colours with the winner receiving a Tag Heuer Aquaracer watch.
The Tigers Player of the Month award is supported by Goldsmiths, the largest prestige and luxury jewellers in the UK, who are also official East Stand sponsors and official timekeepers.