The 32-year-old has only made three appearances during the 2017-18 campaign after falling out of favour at the Bundesliga club.
According to BBC Sport, Compper flew into Scotland on Tuesday and is expected to sign for the Scottish Premiership leaders subject to a successful medical.
Speaking ahead of Wednesday night's clash with Partick Thistle, assistant manager Chris Davies said: "It's not something that is done yet and obviously we're not going to talk too much about a player that is not ours.
"But he is a player who fits the profile for what we see as being a good centre-half. You can see that he is used to a similar game style at Leipzig and other teams he has played at."
Compper, who began his career with Borussia Monchengladbach before five years with 1899 Hoffenheim, joined Leipzig after leaving Fiorentina in 2014.