General:
* Offering
* Proposing
* Tendering
* Putting in a bid
* Making an offer
* Submitting a proposal
In an auction:
* Outbidding
* Overbidding
* Raising the bid
* Increasing the price
* Making a counter-offer
In a negotiation:
* Negotiating
* Bargaining
* Haggling
* Counter-offering
* Reaching an agreement
In a contract:
* Contracting
* Tendering
* Competing
* Submitting a proposal
Other:
* Saying goodbye
* Farewell
* Farewell wishes
* Wishing good luck
* Expressing best wishes
More specific:
* Bidding farewell: Saying goodbye, departing, taking leave.
* Bidding for a contract: Tendering, submitting a proposal, competing.
* Bidding at an auction: Outbidding, overbidding, raising the bid.
The best synonym to use will depend on the specific context and the nuance you want to convey.