Here are my suggested guidelines on an offering price.
First, determine the real value of the property. This can be done by consulting with one or more Realtors and obtaining a CMA (competitive market analysis). That gives you your "Value Number."
Second, if you've been pre-approved or pre-qualified, take that number. That gives you your "Prequalification Number."
Third, determine how much you can comfortably afford to spend, which may well be less than your pre-approved/pre-qualified amount. Figure out what you're willing to spend; how much you can spend without cutting into other investments and living expenses. That gives you your "Comfort Number." In your case, you say that's $1.1 million.
Take those three numbers: Value, Prequalification, and Comfort. Take the lowest of the three. That's the most--the MOST--you should spend for the property. We'll call that Interim Maximum Price Number.
Notice we haven't even discussed the listed price of the house? That's because it has nothing to do with the house's value, what you're qualified for, or what you can comfortably spend.
Now you can look at the listing price. Call that the "Listing Number." Take the Listing Number and the Interim Maximum Price Number. Select the lower of those two numbers. We'll call the results the Final Maximum Price Number. That's the most you can spend. (In your case, it's no more than $1.1 million, and perhaps less, depending on the house's value.) Take the lower of those two numbers. That's your new maximum limit. If the owner's asking for less than the value, less than you're pre-qualified for, and less than you're comfortable spending, then you lower your offer to the listing price.
That's the first part of the process. You now know the maximum you will pay for the property. Your offer should be no higher, and very likely somewhat less, than the Final Maximum Price Number.
There are dozens of different negotiating strategies at this point. And remember: It's not just price. It's also terms and conditions. You might be willing to pay more for a property if, for instance, you need a delayed settlement. You might demand a lower price if you're paying all cash and buying in "as is condition." Those are just examples. Keep the terms in mind, too.
Looking just at price, two often-used strategies are the following:
Split The Difference. The American version of "negotiating" often involves "splitting the difference." Let's say a house is listed for $1.3 million, and your Final Maximum Price Number is $1.1 million. The strategy is to offer $900,000. The sellers may counter at $1.1 million, and you have a deal. Or the sellers counter at $1.2 million. You counter at $1.0 million. And so on. Usually, you end up pretty close to the middle.
Best and Final. You sometimes employ "Best and Final" from a condition of strength--"take it or leave it"--or you do it when "Splitting the Difference" might result in an overly large gap. You make the offer as strong as possible. You provide supporting documentation for your price. You make it clear that it's a "best and final" offer. And you see what happens.
So: Go through the process outlined above. Seek input from your Realtor. Then make an offer.
Hope that helps.