Both words have similar meanings and they both seem phonetically similar as well. So, it seems to me that there's the possibility of some relation between the two words.
I'm curious if one of the words is derived or borrowed from the other, or if perhaps they share some common "ancestor" word borrowed from a completely different language.
According to en.wiktionary.org/wiki/рейс, the Russian
рейс is derived from
reis in Dutch.
And, according to en.wiktionary.org/wiki/reis, the Dutch word appears to have been borrowed from Estonian (at least that was how I vaguely understood the page). And, subsequently the Estonian word was derived from the German
So, my preliminary research has lead me to believe that the Russian
рейс is indirectly derived from the German
Reise. Although, I'm not really confident that this is entirely accurate.