The Guardian say that the club have offered a a four-and-a-half year contract to the 20-year-old who is regarded as one of his country's brightest talents alongside another rumoured Blues target, Breel Embolo. The Daily Mail say the deal would be worth around £3m.
The report suggests that Tarashaj's advisors have been on Merseyside in recent days to discuss the potential move but Grasshopper may insist on loaning the player back for the remainder of the 2015-16 season as part of the fee negotiations.
That wouldn't necessarily square with The Guardian's insinuation that Martinez would allow Steven Naismith to leave in the upcoming transfer window if he can land Tarashaj.
Tarashaj has scored 8 goals in 18 appearances for the Swiss club who are currently second in their league behind Basel. He only signed a new long-term deal in August, however.