For the little that it's worth in the heat of the moment, he had these little periods of 5+ games without a win/draw at Southampton too, but came through them. Unlike Martinez, he's historically been able to turn them around and finish the season where we'd expect him to.
As a club, we probably need answers to two things before deciding Koeman's future though. Firstly, what style of play do we want to play long term? Are there good managers available with a track record of implementing that style and getting results? If there aren't, then we're realistically left with Koeman or Unsworth. You'd then have to question if managing us until the end of the season would help or hinder Unsworth's progression as a coach.
The second question is about what we expect from this season. After the money spent, were we expecting to compete for top 6 and cups? If we were, a change might be a good idea. Were we expecting to finish 7th and not much more? Koeman will probably get us there or thereabouts. Or have we anticipated this to be a transition year with all the new signings? In that case, you could definitely make an argument to just throw in Calvert-Lewin, Lookman, Sandro, Davies, Vlasic, Kenny or whoever else in at the deep end as much as possible and see if they've got what it takes. If that means replacing Koeman, so be it, it might be better for us in the long run if they get games and experience together.