The Reds are currently negotiating a deal which will see Luis Suarez head in the opposite direction, and were hoping to get the Chilean World Cup star thrown into the arrangement.
But the player himself has indicated to Anfield officials that a move to Merseyside does not suit him.
The 25-year-old has been in talks with Liverpool over the past few days, yet he is in favour of a move to London, more specifically Arsenal, with family reasons said to be a major factor in his decision.
The Gunners are ready to move for Sanchez after agreeing a £30m fee with Barca and a contract of £130,000-per-week with the attacker, according to various rumours.
Sanchez was seen as the ideal replacement for Suarez – who netted 31 league goals last season – by Brendan Rodgers, who now looks set to let his star man go for a straight cash deal worth between £70m-£80m.
Liverpool are however pushing through their own transfer plans, with Lazar Markovic set to seal a £20m switch from Benfica in the coming days.