Contact them before you do any work
If you do all the work and THEN contact them and they say no, you now have a game you can't do anything with. A good learning exerciser, but not a very efficient way to run a business if you see yourself as one. If they say no, be mentally prepared to rm -r -f the code and walk away and move on to some other idea.
On the other hand if they say yes to something before you start, then you can proceed and not risk all your work is for naught.
If you ignore their "no", then you are really at risk of legal trouble. The act of asking will put you on their radar big time.
And what if you just don't ask? Maybe they never know, maybe they do but don't care, or maybe they just take you to court. Getting taken to court is like them saying no, with the added fun of you probably owing them something.
Are they a publisher?
See if they accept submissions from outside developers - something a lot of publishers would do by nature of being a publisher. If they do, then present your electronic version of their game as a proposed product. Again, because your proposal is so tied to their product, them saying "no" leaves you with not many options.
One way to approach them would be to ask them if they would be willing to let you do a prototype, with the agreement that they get to make the decision whether to allow it or not. Then at least you've decreased the risk of throwing everything away. If you do this though, make sure that they don't end up owning your demo. You don't want them to say, "No thanks" and then turn around and use your work for free and leverage it to their advantage.
Consider asking to license their content
One way to present the idea of developing the game is to ask them if they would be willing to license their IP for a project. The terms of the license might include money up front, a percentage of income from your work, or a combination of the two. It also no doubt means you will have zero path to using their IP except through them. You're not going to be able to open source your game if you decide to stop working on it for example.
Don't ask them to take risks
When you pitch something to them, make sure it's you taking on the risk, not them. Or at least be willing to equally take on risk. Don't ask them to pay you to develop the game for example. Set it up so that if it does fail, you lose, not them. Someone with strong business and negotiating savvy could probably work that a bit more evenly, but it sounds like you're not bringing a lot to the table. I.e., you're not a studio with 10 years experience and 5 published titles.
Do your homework
Has anyone ever done a version of this game before? Maybe the company did it internally and failed, or it was deemed not a good idea. If so, find out why if you can and learn. Maybe a third party was developing the game in the past and went out of business? Maybe the company just plain hates video games and are board game purists? Get names of people at the company, read their Facebook and Twitter posts, blog posts, and see if there is anything about them on Moby Games. Find out if they ever talk about games and in what way (positive, negative, wishful, cynical, etc.). It's not spying, it's homework that may give you a big hint about how to present things.
Be careful about the impression you create
As an additional comment, be very careful how you present yourself. You sound like you're not yet a game developer, but want to be one. If you tell them that, they may not see you as a viable author. On the other hand if you have examples of existing work to show them, or prototypes of concepts that aren't for release, this may show them you are capable. Have something you can show them as an example of what you can do. Additionally if you are thinking of selling this game yourself (with their permission), be prepared to have a business plan.