The 20-year-old winger, who represents Bulgaria at international level, came through the youth ranks at Craven Cottage and caught the eye during a temporary stint at Colchester United.
Jasper notched eight goals in 16 Premier League 2 appearances for Fulham Under-23s during the 2020/21 season and has made three first-team appearances for the Cottagers.
The London-born forward can operate on either wing or behind the strikers and has plenty of goals and assists to his name for Fulham’s youth teams.
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Shaun Maloney said: “Sylvester is a talented young player who possesses a lot of creativity and speed.
"He gives us extra options and can play on both the left and right in attacking areas. My staff and I are looking forward to working with him and welcoming him to the club.”
Hibs have an option to buy Jasper on a permanent deal in the summer as part of the arrangement.
Meanwhile, Murphy’s time in the Capital has come to an end with confirmation of his loan move to Field Mill, where he will once again team up with Nigel Clough. The 32-year-old attacker played for Clough during a loan spell at Burton Albion during the 2020 season.
The winger, who is out of contract in the summer, made 50 appearances for Hibs during his season and a half in green and white, scoring five times in the process and registering six assists.