Simon, who previously worked for Renault, Ferrari and governing body the FIA, had been with Honda since they announced their return to F1 in 2013.
Simon is on gardening leave from his position and a Honda spokesman confirmed the 58-year-old's contract would end "some time this year".
He refused to comment on reports Simon left as his ideas were being ignored.
Honda has struggled since it returned to F1 with McLaren in 2015 and the Japanese company's programme for this year had a dismal start in last week's first pre-season test.
The company used at least five engines in the course of four days at Spain's Circuit de Barcelona-Catalunya as a series of reliability problems restricted running.
Honda F1 boss Yusuke Hasegawa admitted he "worried" about whether the company could solve a problem that led to an engine failure before the start of the season in Australia on 24-26 March.
And McLaren racing director Eric Boullier admitted that the situation was "not good enough".
Boullier added: "There is a bit more work to be done in Japan to investigate why we had those issues: issues that we absolutely did not expect to have - and for sure neither did Honda.
"It is fixable, but it was not the plan to have these issues. They are not really serious, as there are no fundamental issues with the design."
French engine expert Gilles Simon has left his role as a consultant to Honda's struggling Formula 1 programme.