He later apologised and removed the Tweets, writing "When you are sick. You are sick. Jumping to conclusions helps nobody. My mistake @FabioAru1. I should shut my mouth. Sincere apologies."
While Aru had previously said he planned to take Henderson to court, the New Zealander had hoped the incident had died away after he apologised.
However, Cyclingnews.com has reported that Aru's agent Alex Carera and lawyer Giuseppe Napoleone had struggled to get proceedings underway because Henderson is based in Spain but rides for Belgium team Lotto Soudal.
Legal action will be taken "to protect the good name and honour of Fabio Aru and the team [Astana] that he rides for," Cyclingnews.com said.
Henderson said on Tuesday that he still had not heard anything from Aru's representatives.
Both riders are currently involved in the first Grand Tour of the season, the Giro d'Italia.