Carlo Ancelotti and Jurgen Klopp are both regarded as favourites to take over from the Northern Irishman.
No sooner had his team earned a point against Everton at Goodison Park in the 225th Merseyside derby, Rodgers was sacked by owners Fenway Sports Group.
He received his marching orders via a trans-atlantic phone call just 90 minutes after the 1-1 draw against the Toffees.
Adam Lallana said in an Instagram post: “Thank you Brendan for bringing me to such a fantastic club. Also helping me fit in when I first arrived. Good luck #LFC #YNWA.”
Goalkeeper Simon Mignolet, who was signed by Rodgers in 2013 for a fee of £9million, uploaded an image to Instagram of himself and the former boss with the hashtags “#ThankyouBrendan #goodluck #YNWA.”
Liverpool hope to have named Rodgers’ successor before their next Premier League game against Tottenham on October 17.
Daniel Sturridge continues to be blighted by injury and Liverpool are keen on bringing in Higuain as an insurance policy.
Higuain, 27, was linked with a move to Arsenal but manager Arsene Wenger decided against a move, instead opting to bring in goalkeeper Petr Cech from Chelsea for around £11million.
The South American forward, moved to Italy after seven years in Spain and scored 18 goals in 37 games last season in Serie A.
Written by Jack Wynne – @JackRobertWynne