You use the word “offre”/offer in your question, but in your example (between spouses), I see the exchange more as a “request/order” followed by a positive “response/acquiescence” thereto than one of a “solicitation of an offer” followed by an “offer.”
With this interpretation of your example in mind, I would agree with the other answers that “Moi, je vais le faire” would be the most idiomatic and appropriate form to use.
However, in an arm’s length transaction, where an offer has to be accepted to seal the deal, I think that “Moi, je vais le faire” would be at best a bit presumptuous on the part of the maker of the “offer” in response to a real solicitation for offers.
Of the two choices, I think a lawn service technician, for example, would say, regardless of how soon s/he would begin the work, “Moi, je le ferai [pour $75.00 si vous êtes d’accord]” more often than “Moi, je vais le faire [pour $75.00 si vous êtes d’accord ou pas]” to avoid sounding too presumptuous.
Even between the related parties in your example, Marie might have a very legitimate reason for not wanting to accept Jacques’ offer (perhaps he mowed her prized Tulips the last time he did it and she’s still skeptical, for example), and in that case “Moi, je le ferai [si tu veux, ma Chérie]” would seem to be a bit less presumptuous than “Moi, je vais le faire [si t’es d’accord ou pas]” and be closer to a “real” offer (i.e., one that can be either accepted or refused).
Regardless of the validity of the possible “presumptuous” nuance that I see between the two choices, I would personally find another way to make a real offer and I’d avoid the use of the future altogether (simple ou proche) of “faire.”
Using “pouvoir” would work, I think, to put less pressure on Marie to accept Jacques’ offer: “Je peux le faire [si tu veux].” or even “Puis-je t'aider ?”
The use of the imperfect tense of “faire” -- "Et si je le faisais?" -- would be another way to suggest an offer (or at least to offer a suggestion) and avoid a possible repeat of the "tulip incident."