Be careful.
Listing a property to the open market is the only definitive way to *know* how much a house is worth. Everything else is projections and guesses. So, if you sell privately, you must must must be an absolute expert in your market.
1: If you got the 1.3 and 1.4 numbers from your agent, all-but ignore them. It's not fair to paint all agents with the same brush, but many agents will set low exceptions so that they can easily clear them.
2: The investor putting an offer on the house will likely know the market better than the buyer. It's that investor's job to know the market, and squeeze deals and money out of it. In this case, if I'm the buyer, I'm following the steps described if the house is worth 1.4 - 1.8. I'm probably NOT acting that way if the house is worth less than 1.4.
3: Feeling a sense of urgency is a huge risk in negotiations. When you get the feeling in the pit of your stomach "oh no - this one might get away" - you're already in a loosing position.
For those reasons, assuming that your fees are the same regardless of whether you list it or not, I would personally lean towards listing it. There's nothing wrong with making a deal here, a 1.4+ cash offer sounds great, but there's simply too much risk that I'm leaving money on the table. So, I'd list, but that's obviously a risk in itself.
With that said, DO NOT ACCEPT THAT OFFER. If you want to make a deal with this person, counter!!! If the process is bid, reject, higher bid - the correct next step is for you to say "I'm happy to make a deal with you, but if we want to make this happen before listing it on the MLS, I'd need to get to 1.5M." If they say no (and they might), there's a great chance that you can still make the deal at what they have already offered.
Happy hunting, and let us know how it turns out.