Sanchez has scored 13 goals in 24 appearances for Arsenal since moving to the Emirates Stadium from Barcelona in a deal worth a reported £35million.
Yet Rodgers, whose side host Arsenal on Sunday, was eager to bring the Chile international to Anfield as a replacement for Luis Suarez.
Liverpool have struggled for goals this season, with the absence of Daniel Sturridge due to a string of injuries coming at a particularly inconvenient time following Suarez’s departure to Barcelona.
Discussing his club’s attempt to bring in Sanchez, Rodgers said: “All I know is that he’s a world class player. He was identified as someone who could come in and be perfect for us.
“He’s a brilliant player with outstanding quality and even bigger work rate.