Bid is also used in describing sports (again, a context of competition). 'The Green Bay Packers made a last-minute bid to tie the score, but failed.'
'Offer' can also be an attempt, but it is used in different contexts. You might, as an individual buyer, make an offer to buy a used car at a certain price, for instance, as a starting point to negotiate the price or haggle directly with the owner. No other buyers are involved.
A business might offer you a job, hoping that you'll work for them.
You might also offer to help someone who appeared to need help.
Usually, there a clear possibility for a bid to be rejected or outbid. Someone making an offer, on the other hand, usually expects that it will probably be accepted, or at least that it will result in further negotiations until a satisfactory arrangement is reached. It is certainly possible, however, for someone to decline (more politely) or outright reject (less politely) an offer.