Can other users see close votes on my question if they haven't reached the threshold to close? If so, does this not bias users to vote to close more often?
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Yes those with rep to close can, and it might.
However remember something important, even the people with close privileges aren't perfect, and seeing what the community thinks can help them learn what is or isn't acceptable.
As for the bias part, I would hope that those who do have this privilege would close/reopen the question on the merits, not because, "Oh, this question needs only one more vote to close it."
As the other answers state, those with the vote to close privilege can see the number of close votes a question has attracted and a breakdown of which reasons have been given.
However what hasn't been mentioned thus far is that you can see who has voted to close a question if that vote was cast from the close votes queue, for example on a currently still open question:
Where everyone with a "close" remark has voted to close the question from the review queue. This is another edge case in a similar category to the automatic duplicate comment (though this will probably be removed very soon).
There is an entire Close Votes review queue, where users with at least 3k reputation can see questions that have either pending close votes or it doesn't belong here flags.
Once you click "close" in either the question or the review queue, you can see how many users have voted to close for a specific reason.
This should not bias others to vote to close. The queue itself has a Leave Open button; if a sufficient number of users press it, the question disappears from the queue.
Note that there's also a Reopen Votes review queue.
In addition, 10k users can see those and some other pending flags inside the moderator tools. They can either vote to close or cast an invalid flag flag.
Without those tools, moderating the currently 4,500,000 question on Stack Overflow would be impossible. The Close Votes queue alone holds almost 50,000 questions right now.