The phrase "make a bid for" means to attempt to obtain or acquire something, often through a competitive process. There are several synonyms that can be used to convey a similar meaning, such as "place an offer for," "put in a proposal for," "attempt to secure," "vie for," "compete for," "pursue," and "try to attain." These synonyms can be used interchangeably depending on the context and the desired effect. They all signify a determined effort to obtain something, whether it be a job, a contract, or a prize. Using such synonyms can add variety and depth to one's language and help to better convey the intended message.