I agree with Callum. I'm not sure when "release the offer" would ever be used actually. Callum was saying that "release the offer" could be used when this offer is first given, but i think that would be closer to "launch the offer" or "launching the advertising campaign" (the ad campaign would have the offer in it of course) according to your examples ALTH you should definitely be using "extend" and not "release". I can't even think of a situation where you would need to use the "release the _____" . . . unless it's the KRAKEN

'Extend the offer' talks about giving an opportunity to someone, as seen in your example. It can also refer to making the duration of an offer longer. For example, a shop might 'extend the offer' of half-price products for another week. 'Release the offer' is perhaps less commonly used. It might be used to talk about when the offer is first given, or first 'offered'. Therefore, the two phrases are not exactly the same, but their meanings can overlap.

‘Release’ isn’t correct. ‘Extend’ is good but fairly formal. We decided to make the offer to Sophie. (Less formal) We decided to offer the job/position/post to Sophie.

Thank you