I disagree with most of the answers here.
Don't sell the logo.
It just sets you up for a lot of speculative work in the worst of ways. If they do accept it, any customers they refer will expect you to first do some work, or even a lot, before they decide to hire you.
However, since the logo is already made, you've got a rather bad bargaining position. Either you take what they offer, or you make no money.
Go big, medium or home.
You current plan can either go medium, or home.Instead of having a 100% finished project you try to sell, with the chance at it backfiring, up the ante. Instead, treat it as a 10% finished project, and try to sell that. If they accept, you go big, you get a lot more work. If they say, no thanks, but we do like the logo, go medium, and you still get something out of it. If they decline entirely, you go home.
You've now got an additional, far better, possible outcome, and you've raised the initial negotiation perspective.
Show the logo, offer more.
Show them the logo, mention your awards, ask what they think of it, and if they would be interested in having you fresh up their branding. Don't mention selling/giving the logo on its own; if they don't want to hire you or offer you money for the logo out of their own accord, the logo stays copyrighted to you.
To keep my portfolio filled with fresh new work, I occasionally take
part in design challenges. For one of these, I recently revamped the logo of [yourcompany]. It's gotten a lot of positive responses and several awards; award1, award2, and award3.
If you're interested, we could have a look at refreshing the rest of [yourcompany] branding, and perhaps get the company some extra media attention.
Try not to ask them to buy something, instead offer them an opportunity.
Negotiation and numbers
If they're interested and you're getting down to numbers, look at how long the logo took and how much your hourly rates are. That's how much the logo would cost, normally. Then offer them,if they agree on hiring you for the brand refresh, you throw in the logo for free / as a discount.