In Romanian language, there is an expression that would be translated to English as "I offer and request seriousness", that is usually used for example at the end of an offer/ad. What is the English equivalent for this expression? I thought about "Please consider the seriousness of this offer" but I'm not sure about the correctness of this. Also, what alternatives can I use?
Avoid the entire idiom.
Anything asking me to consider the degree to which an offer is serious could just be (as in your case) because English isn't their first language, but I get a strong vibe that someone is trying to strong-arm me (it sounds like the sort of thing a lawyer would say when their client hasn't a good case).
You may mean "please consider this offer seriously" - in that it is I who should be serious, not the offer.
Otherwise, just don't tell me that the offer is serious. If it wasn't serious I would assume you wouldn't be making it, so "please consider this offer" would suffice.
Indeed, if you've made an offer, do you really need to tell me to consider it? Isn't that implied? Maybe just skip it entirely.
The phrases "in good faith" and "good faith offer" are legal terms used in distinguishing real offers from false ones. Book links at ngrams for good faith offer,in good faith include sentences like
Thus, some definition of a good faith offer to purchase should be included in every restriction. ... The agreement may provide that the good faith offer must be supported by an earnest-money deposit.
from The law of business organizations by John Moye, and
The good faith offer retroactively erased the fornication. Before Nesbitt could reply, the district attorney objected, insisting that the court allow the state to test Wright's good faith in offering matrimony.
from Ardor in the Court!: Sex and the Law by Jeffrey Miller.
I offer these terms not as suggestions for use in making offers, but merely as examples of ways to refer to serious offers vs those not serious.
"Please consider the seriousness of this offer" sounds like there are some unspoken dire consequences to not accepting the offer. If this is not that kind of offer, as I suspect, then one might follow up an offer with a simple "think about it", to push it a little more. Other neutral and simple phrases can be used but, any phrases with connotations of cajoling will be rejected outright. Being overly timid isn't good either. A few years ago a overseas car manufacturer ran a long ad campaign here with the slogan "please consider". The campaign didn't go down very well.