Learn English – Very keen to offer

The formal (financial) usage in these contexts is "tender offer." This basically means "bid." "Tender" is the adjective and "offer" is the noun.

But "tender offer" is often abbreviated to "tender." So the billionaire won his "tender" or bid to build the Moscow St. Petersburg Highway.

C. John suggested helping Jane with the washing-up.

Consider some contexts in which this sentence might occur.

Julie asked John what she could do to speed things up. John suggested helping Jane with the washing-up.

Jane pointed out that Julie was old enough now to have some regular chores, and asked John if he had any ideas. John suggested helping Jane with the washing-up.

suggest VERBing does not imply that VERB has any particular subject; specifically, it does not imply that the subject suggests that he should VERB. Consequently, this sentence does not, as the quotation does, imply that John made an offer to help; it at most puts helping Jane forward as a good idea.