Brendan Rodgers sent scouts to watch the Swedish winger on a number of occasions last season but were priced out of a move.
He was seen as a potential replacement for Patrick Roberts, who has since returned to Manchester City after his loan spell expired.
However, the Dutch club were demanding close to €9million for their prized asset even though he has just the 12-months remaining on his current deal.
0-1 Sam Larsson Goal International Friendly… by milena-georgieva1
The latter is believed to be leading the race to sign the versatile forward and a deal could be completed over the weekend.
La Viola are in the market for a winger following the sale of Federico Bernardeschi to Juventus. They passed up on the opportunity to sign Liverpool attacker Lazar Markovic for £15million and have focused on Larsson instead.
Stefano Pioli’s men have tabled a €6million bid and the feeling is the player will push for the move.
However, Football Whispers understands Torino have a strong interest in the winger but are yet to table an offer. And time is running out.
Il Toro had hoped to bolster their attacking options this summer but questions marks surrounding AC Milan transfer target Andrea Belotti’s future has made it difficult for Siniša Mihajlović to do business.