The four-time Tour de France winner, who denies breaking any rules, could potentially be stripped of his Vuelta a Espana title after a urine test during the race in September had shown excessive levels of Salbutamol.
Salbutamol is permitted as a legal asthma drug by the World Anti-Doping Agency (WADA).
The UCI said Froome's failed urine test did not necessitate a mandatory provisional suspension even if analysis of his urine sample revealed a concentration of 2000 nanograms per millilitre (ng/ml), double WADA's threshold.
"Team Sky should suspend Froome," Lappartient told daily Le Telegramme.
"However, it is not up to me to interfere. Without going into the question of guilt, it would be simpler for everyone," said the Frenchman, who was elected last September.
"It's up to (team manager Dave) Brailsford to take his responsibilities. Apart from that, I think that it is what the other riders wish."
Lappartient, who said he was notified of Froome's test result an hour after being elected in Bergen, Norway, believed that "in the eyes of the wider public (the Briton) is already guilty".
Frenchman Romain Bardet, who finished on the Tour podium in the past two editions, said earlier this week Froome should pull out of racing until the case was over.
The affair, however, is likely to drag on, according to Lappartient.
"It's going to be a long judicial battle. It can last at least a year," he said.
Meanwhile, world champion Peter Sagan fended off a question about the Froome controversy after he rode superbly to win stage four at the Tour Down Under and take the race lead, saying he could not comment.
"Well, I can't respond - (we are) thoroughly different riders," Sagan said. "I've never ridden with him on the same team and then we will see what they (the authorities) are going to decide."
Bora-Hansgrohe media managers shut down further questions about the issue, highlighting the sensitivity around the ongoing investigation.