Is the request by a government official to a private company to provide services or purchase furniture for the department, in exchange for the company winning a bid to train the department's employees, considered a bribe, especially since the value of the offer can be increased to cover the costs of these services?

The forbidden bribe is that which is given to nullify a right or establish a falsehood. As for what is given to obtain a right or a deserved benefit that cannot be accessed except by offering something, its sin is upon the recipient, not the giver. So, if your act of purchasing furniture, printing a guide, or setting up a computer network for the administration aims to obtain a legitimate right for you, there is no sin upon you, and the sin is upon the officials who stipulated it. However, if this act is intended to nullify the rights of other companies to obtain training, then this is a forbidden bribe for both parties, because it nullifies a right and establishes a falsehood.