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?

I would add to the excellent answer of Jeremy Miles by saying that how you treat your p-values also strongly depends on what you want to do with them. They get a bad reputation (and rightfully so), for being the deciding factor between "Your work is worth publishing" and "Your work is garbage".
However, what you take from your p-value depends for instance on your evaluation on how much you want to avoid type I / II errors.
Let's assume you're running a large-scale clinical trial for very cancer medication which is very expensive and heavy on side effects, and in the end you test whether the treated group had better survivability than the control group. By the medical context, you are incentivised to reject the null hypothesis "The medication is no better than placebo" then and only then if you are very certain that the medication is beneficial. Something like "our p-value is 0.08 which isn't significant but look, there's a trend" isn't going to cut it there.
If you take another example, and you are trying to sift through a large dataset to find associations between, say, environmental factors and microbiome, and you find an anticorrelation between PM2.5 pollution and the abundance of Parabacteroides golsteinii. Then with your statistics you're usually not laying claim to any clear conclusions, but are simply saying "Hey, this could be worth investigating further", in other words, your main goal is hypothesis generation. If here you end up with $p=0.08$ (especially if you fastidiously corrected for multiple testing), it could still be worth mentioning that sure, it wasn't significant at the $\alpha = 0.05$ level, but maybe we should have a look. (And in a perfect world, someone interested in this would then design a dedicated study to see whether there is a link or not).
In other words, if we look at a p-value not as a magic number, but a measure of effect size, or the weight of evidence, it can be seen in a more nuanced way (including for example interpreting it while taking into account other existing evidence on a hypothesis).
