TL;DR: Bribes that pervert justice are condemned, but gifts which curry favour are allowed and sometimes even a sign of wisdom.

שׁחד

The Hebrew word for 'bribe' in these verses is the root שׁחד. It only occurs 25 times in the Hebrew Bible which means we can do a comprehensive word study pretty easily. Below I will list most of the verses with this root.

I think the data shows the word is used in two contexts. I'm not sure if the word therefore has two distinct senses. I think it might really just have one sense, perhaps a 'gift of enticement' would convey the meaning well, which can be used in just and unjust contexts.

Bribes which pervert justice are condemned

The most common context is when a gift is given to pervert justice. Here are the texts (from the ESV) condemning such bribes:

Exodus 23:6-8: You shall not pervert the justice due to your poor in his lawsuit. 7 Keep far from a false charge, and do not kill the innocent and righteous, for I will not acquit the wicked. 8 And you shall take no bribe, for a bribe blinds the clear-sighted and subverts the cause of those who are in the right.

Deut 10:17-18: For the LORD your God is God of gods and Lord of lords, the great, the mighty, and the awesome God, who is not partial and takes no bribe. 18 He executes justice for the fatherless and the widow, and loves the sojourner, giving him food and clothing.

Deut 16:18-20: You shall appoint judges and officers in all your towns that the LORD your God is giving you, according to your tribes, and they shall judge the people with righteous judgment. 19 You shall not pervert justice. You shall not show partiality, and you shall not accept a bribe, for a bribe blinds the eyes of the wise and subverts the cause of the righteous. 20 Justice, and only justice, you shall follow, that you may live and inherit the land that the LORD your God is giving you.

Deut 27:25: Cursed be anyone who takes a bribe to shed innocent blood.’ And all the people shall say, ‘Amen.’

Isaiah 5:23-24: Woe to those who are heroes at drinking wine,
  and valiant men in mixing strong drink,
23 who acquit the guilty for a bribe,
  and deprive the innocent of his right!

Psalm 15:1, 5: O LORD, who shall sojourn in your tent?
  Who shall dwell on your holy hill?
5 [He] who does not put out his money at interest
  and does not take a bribe against the innocent.
He who does these things shall never be moved.

Psalm 26:9-10: Do not sweep my soul away with sinners,
  nor my life with bloodthirsty men,
10 in whose hands are evil devices,
  and whose right hands are full of bribes.

Job 15:34-35: For the company of the godless is barren,
  and fire consumes the tents of bribery.
35 They conceive trouble and give birth to evil,
  and their womb prepares deceit.”

Proverbs 17:23: The wicked accepts a bribe in secret
  to pervert the ways of justice.

2 Chron 19:7: Now then, let the fear of the LORD be upon you. Be careful what you do, for there is no injustice with the LORD our God, or partiality or taking bribes.”

And we have some examples:

1 Samuel 8:3: Yet his [Samuel's] sons did not walk in his ways but turned aside after gain. They took bribes and perverted justice.

Ezekiel 22:12: In you [Jerusalem] they take bribes to shed blood; you take interest and profit and make gain of your neighbors by extortion; but me you have forgotten, declares the Lord GOD.

Micah 3:11a: Its [Israel's] heads give judgment for a bribe;
  its priests teach for a price;
  its prophets practice divination for money;

So what we would call bribes are completely forbidden. You can't give a bribe to escape justice or a punishment, to induce a judge to wrongly convict someone else. The tender process probably didn't exist back then, but I think it's clear these verses would condemn the use of bribes to cheat and wrongfully win contracts as well.

Gifts to gain the favour of another

The other context is much rarer. There are only two verses endorsing such gifts:

Proverbs 17:8: A bribe is like a magic stone in the eyes of the one who gives it;
  wherever he turns he prospers.

Proverbs 21:14: A gift in secret averts anger,
  and a concealed bribe, strong wrath.

But we do see some examples of people giving such gifts. They're not positive examples though, such as these two kings who gave the treasures of the temple to other nations:

1 Kings 15:18-19: Then Asa took all the silver and the gold that were left in the treasures of the house of the LORD and the treasures of the king’s house and gave them into the hands of his servants. And King Asa sent them to Ben-hadad the son of Tabrimmon, the son of Hezion, king of Syria, who lived in Damascus, saying, 19 “Let there be a covenant between me and you, as there was between my father and your father. Behold, I am sending to you a present of silver and gold. Go, break your covenant with Baasha king of Israel, that he may withdraw from me.”

2 Kings 16:8: Ahaz also took the silver and gold that was found in the house of the LORD and in the treasures of the king’s house and sent a present to the king of Assyria.

It was wrong to take the treasures of God to give to another, and it's wrong to trust in your own means rather than God, but these examples are more misuses of an allowable option than proof the option of giving gifts is categorically as wrong as giving bribes.

Ezekiel likens Israel's idolatry to a prostitute who pays/"bribes" its customers to sleep with her. And Proverbs 6:35 says that you can't make up for sleeping with someone else's wife by giving them gifts.

Ezekiel 16:32-34: Adulterous wife, who receives strangers instead of her husband! 33 Men give gifts to all prostitutes, but you gave your gifts to all your lovers, bribing them to come to you from every side with your whorings. 34 So you were different from other women in your whorings. No one solicited you to play the who*e, and you gave payment, while no payment was given to you; therefore you were different.

Proverbs 6:35: He [the jealous husband of the woman you commit adultery with] will accept no compensation;
  he will refuse though you multiply gifts.

I think the kind of gifts which are allowed and even considered wise would be gifts like giving a bottle of wine to a new client, or taking your favourite clients out to a fancy restaurant. Even secret gifts can still soften anger, such as when countries trade spies with each other. But this is a matter of wisdom, and to ensure that gifts like these are not actually bribes, many jurisdictions now require those in public office to declare all gifts, and they may put limits on what can be accepted. We should consider carefully what is communicated by such gifts in our present culture.