Teenage fly-half Heathcote, who has made 13 appearances and scored 78 points since making his debut against Gloucester this season, has signed a three-year contract to remain at the Rec.
Prop Palma-Newport, who spent a spell on loan at Leeds Carnegie, has committed to a two-year deal.
Director of rugby Sir Ian McGeechan said: "We are delighted that Tom and Kane have both chosen to stay with the club.
"We've altered the way we operate the academy this season, integrating the young players with the first team much earlier and by having them train with the senior squad full-time this has had real benefits, both for the players themselves and the team.
"Both Tom and Kane are fantastic young talents, who have shown great maturity and ability already, and we are sure they will continue to do so as they develop and grow as players."
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