The Bhoys failed to win their fourth league game in a row on Wednesday night when they only managed a 2-2 draw at Livingston.
The current run is Celtic’s worst in the league without a win in 21 years and they are 20 points behind Rangers in the league table having played two games fewer than their Glasgow rivals.
Some fans have called for Lennon to be sacked, but the Celtic board continue to stand by him and it is unclear if and when a change will be made in the dugout.
Hartson feels if the Celtic manager is allowed to bring in players in the next two weeks it will be a signal that he could still stick around at Parkhead.
If Celtic do not act, Hartson thinks that could be a sign that Lennon will go, while he also cast doubt on whether it is worth making signings this month given the club’s league situation.
Hartson said on the Ten10 Celtic Podcast: “I think January will tell us a lot, because if they are thinking of changing the manager then Neil won’t make any signings between now and the end of January.
“They are not going to give Neil Lennon any money if they are thinking of making a change.
“If Neil goes and buys four players this week then that will tell you a totally different story. That would show me Celtic are going to keep Neil on and that they believe in him still.
“So I think the next ten days tells us.
“There are players out there, we need defenders, we need strikers, we need to be strong.
“But is it worth going in the market now when we could be possibly 23 points behind by Monday next week?”
Some of Lennon’s big signings from the summer, such as Shane Duffy, Albian Ajeti and Vasilis Barkas have not yet found success at the club.