Interpretation of p-value near alpha level If the alpha is set to 0.05 I have encountered many scientific publications saying that a "tendency for an effect" is present based on a p-value of 0.05 < p-value < 0.1. On the other hand, I have had statisticians who criticize me for doing so, because there is only "reject" or "not reject". Therefore, it does not make sense to distinguish between a p-value of 0.08 or 0.97. Also, some statisticians have criticized me for reporting p-values as p<0.05 because it is not precise.
My question is: how to deal with a p-values that are not below but close to my alpha?

There are two different approaches to interpreting statistical significance - the Fisher way, and the Neyman-Pearson way. We smush these together (into what Gerd Gigerenzer has called a 'bastardised approach'). The reason that statistical significance testing [Edit, n italics] as it is often taught and discussed doesn't seem to make sense is that, essentially, it doesn't.
Neyman-Pearson said that you pick a cutoff and you use it. It's less than the cutoff(say, 0.05) or it's not. There's no other information to convey.In NP, 0.08 and 0.97 are the same.
Fisher said you take the p-value and you treat it as the level of evidence that there is an effect. <0.2 is some evidence, but it's pretty weak; <0.1 is a bit better, but still kind of weak. <0.05 is what Fisher said is often good enough (but he also wrote that one should change one's significance level according to the situation, which no one does).
Either report the exact significance level and interpret that appropriately. Or use 0.05. Don't do this nonsense of 0.10>p>0.05.
Your p-value presents some evidence. It's not great evidence, but it's not no evidence. You shouldn't be trying to say "Yes" or "No" when maybe is an answer.
In addition, people often say that their p-value of 0.06 is "approaching statistical significance". No one says it is "Running away from significance" or that their p-value of 0.04 is "Approaching non-significance."
