In Monopoly, I can sell an unimproved property (mortgaged or non-mortgaged) to another player at any agreed price. Instead of negotiating the price with individual players, can I just conduct an auction?
From the rules:
SELLING PROPERTY...Unimproved properties, railroads and utilities (but not buildings) may be sold to any player as a private transaction for any amount the owner can get; however, no property can be sold to another player if buildings are standing on any properties of that color-group. Any buildings so located must be sold back to the Bank before the owner can sell any property of that color-group. Houses and hotels may be sold back to the Bank at any time for one-half the price paid for them. All houses on one color-group must be sold one by one, evenly, in reverse of the manner in which they were erected. All hotels on one color-group may be sold at once, or they may be sold one house at a time (one hotel equals five houses), evenly, in reverse of the manner in which they were erected.
You can see that it can be sold to any player as a private transaction. There is nothing in the rules that forbid an auction.
Strictly by the rules, no. Auctions only occur if a player lands on an un-owned property and chooses to not buy it for its listed price.
However, as you say, unimproved properties can be sold to any other player for any agreed price. The rules don't say how you come to agree on which player, or what price, so you could always ask the other players what they'd be willing to pay for, say, Mediterranean Avenue, and sell to the highest bidder. It wouldn't be a binding deal, though, given its exclusion from the rules (so if the highest bidder isn't the player, or amount, you're looking for, you're not bound to sell to them).
There are several different levels on which this question can be answered. First, there is no mechanism given in the rules for holding such an auction. So it is not possible to hold such an auction in the game.
But just because you can't perform an activity as a game action doesn't mean that you can't do it. There's nothing in the rules allowing you make a sandwich while you play the game, but that doesn't mean you can't do so; the making sandwich simply won't exist as something that is happening in the game. So you can engage in an auction as a non-game activity you are doing while playing the game, and then engage in the in-game action of selling the property based on the auction. And in fact, attempts to sell a property can naturally turn into an auction; if one player proposes to buy the property, another player will likely interrupt to offer a larger price, and so on.
But on a third level, auctions tend to be complicated, time consuming, and, at least to my tastes, boring affairs. Depending on your playing group, they might get annoyed at you for interrupting the game and making them wait while you engage in side activities before finishing your turn.
So, in conclusion: yes, but if you want to do it, you should have a discussion with the other players about what house rules they want regarding it.