In my regression, there are 4 independent variables. But, the constant still comes at negative reading if I remove 2 of the independent variables. The variables are Ln converted.
First, I'm assuming that you are using OLS (ordinary least squares) regression.
Second, the constant/intercept in OLS regression can conceptually be zero if zero is the best estimate for scores on the DV when all of the IVs are zero. Whether that makes sense for your situation depends on the variables in your model, your sample, and factors such as whether mean-centered (or some other adjustment) scores were used for your IVs.
The test of the intercept/constant in a regression equation merely informs as to whether the hypothesis that B0 = 0 is reasonable or not for the linear model. The p-value is a function of: (a) the value of the estimated intercept/constant; (b) the total number of IVs in the model; and (c) the sample size.
I'm not sure from your query whether you're concerned about the point estimate for B0 being negative, being non-significant (depending on your alpha level), or both.