Is it permissible to offer a bribe in the form of a gift (a mobile phone) to a purchasing employee who caused damage to the work, in order to دفع ضرره وإحقاق الحق [prevent further harm and establish justice], and what is the ruling on someone who does so based on their company's request?

The basic principle is that bribery is forbidden and considered among the major sins, due to the Prophet's (peace be upon him) saying: "The CURSE of Allah is upon the one who gives a bribe and the one who takes it," as well as the intermediary between them. It is forbidden to demand, accept, or offer a bribe. However, this prohibition is specific to bribes used to establish falsehood, invalidate truth, or take what one is not entitled to.

As for a bribe used by a person to attain their right, or to repel injustice or harm, it is permissible according to the preponderant opinion. In this case, the sin falls upon the recipient (the taker), not upon the briber (the giver). If it is possible to deter the employee from their action, then it is not permissible to pay the bribe. However, if that is not possible due to the collusion of officials with them, and your institution deserves the tender, then it may be permissible to pay it, and the sin will be upon the taker.