The Daily Mail claims both clubs are keen on the pacy 24-year-old, who has stood out for the Swiss league leaders.
Assale has scored 18 goals this season for the Bern-based club.
Saints have a permanent scout in Switzerland and have checked on Assale, who is keen to make the next step up in his career, several times.
But Leicester City have also been over to watch the Ivory Coast forward with Young Boys now hoping an auction breaks out between the two Premier League clubs for their £12million-rated asset.
Assale is only 5ft 6ins tall but is a cool finisher and has provided over 11 assists for teammates this season.
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