Tynecastle officials put a strong pitch to the English Premier League club in the hope of negotiating a deal for the 21-year-old, who scored seven goals in 20 appearances in Scotland last season.
He became a popular figure among Hearts supporters with important strikes both in the Premiership and during the club’s run to the Scottish Cup final.
Everton are aware that the Edinburgh club want Simms to return north but they are unlikely to make a final decision on his future until well into July, and potentially even August.
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A number of teams in the English Championship are interested in the player as a result of his productive period at Tynecastle. Everton are interested in loans but a permanent transfer has not been ruled out.
Manager Frank Lampard will assess Simms during pre-season before deciding whether he should remain part of the first-team squad at Goodison Park or head out on loan.
With fellow strikers Dominic Calvert-Lewin and Richarlison attracting attention from other clubs, any sale could mean Simms is needed until Everton can recruit suitable replacements.
Hearts hope the attraction of European football next season works in their favour if he is to be loaned out. They are guaranteed group stage matches in either the Europa League or Europa Conference League and the Englishman would be an asset to their cause at that level.