Is it possible to use linear model $Y=aX_1+b(X_1\times X_2)+c$? I am doing linear regression with SPSS. There are two potential Predictors $X_1$ and $X_2$ for Dependent Variable $Y$. I found that $X_1$ and $X_2$ are strongly correlated $(r=0.63, p<0.001)$. So I induce the interaction term $X_1*X_2$. However, I found the Model 1, 
$Y=a_1X_1+a_2X_2+b(X_1\times X_2)+c$, $(\text{sig.}=0.098)$ 
is less significant than Model 2, 
$Y=aX_1+b(X_1\times X_2)+c$,  $(\text{sig.}=0.040)$, 
and the standardized coefficients of $X_2$ in Model 1 are near $0$ $(\text{sig.}=1.000)$. Since $X_1$ is the main predictor I am interested, could I use Model 2 instead of Model 1?
 A: *

*It is definitely possible to fit a linear model of the form $Y=ax_1+b(x_1\times x_2)+c+\epsilon$ (don't forget the error term)

*However, it may not make sense to do so.
In particular your reasoning doesn't seem to make sense to me: 

I found that X1 and X2 are strongly correlated (r=0.63, p<0.001). So I induce the interaction term X1*X2.



*

*Use of an interaction term is not justified by the IV's being correlated. The two things are not particularly connected.

*If you are fitting an interaction, it would be quite unusual to omit either main effect.
A: No.  It is very rarely a good idea to include an interaction term without including all of the main effects. The resulting model will almost always be nonsensical. And you shouldn't judge your models on significance. 
A: With your two predictors being highly correlated, you might have a problem with multicollinearity. Have you tried running a ridge regression? I'm also wondering if your two predictors are so highly correlated that--instead of an interaction between the two predictors--you could look at a quadratic trend of $X_1$? That is:
$y = \beta_0 + \beta_1 X_1 + \beta_2 X_1^2 + \epsilon$
