The 35-year-old is having a medical at the Scottish club, who are now managed by his former Reds boss Brendan Rodgers, according to Sky Sports.
The Ivory Coast international defender, who has 118 caps, should be available for Wednesday's Champions League qualifier against FC Astana - despite the deadline for signings being July 21 - as clubs are allowed to have a single transfer up to a day before kick-off.
Rodgers praise 'infrangibles' as Celtic finish season unbeaten
Toure was released at the end of last season, despite playing 26 times for Liverpool, and would become Rodgers' second summer signing after Moussa Dembele.
He began his career at Arsenal in 2002 and won the 2004 Premier League title before joining Manchester City in 2009 and then Liverpool in 2013.
Unbeaten Celtic end astonishing season as Invincibles with win over Hearts
Celtic one win away from Invincibles status