This now becomes effectively an "errors in variables" problem. Nothing stops you from trying to estimate the parameters using traditional methods. Essentially, the curve fitting tools or optimization TB tools, stats tools, etc., do not explicitly force you to say where the error comes in. (There are implicit assumptions with all of these tools.)
You will merely fit the model as:
The only problem that will arise is that you may get somewhat biased estimates for the parameters. That is at least the classical problem when you have errors in the independent variable. Your ability to estimate the parameters well will depend on the magnitude of that noise, compared to the variations in X.