In his response to this question, Dr. Monzer Kahf, a prominent economist and counselor states:

“From the point of view of permissibility, it is always permissible for the questioner to go for Hajj while under a Loan, if it is without interest and if he can afford to spend on Hajj without negatively affecting his ability to pay the loan back.

Interest-based loan, if not taken out of necessity that relaxes the prohibition, is one of the most grave sins from Shari’ah point of view and it is always much better to rid oneself of sinful conditions before going for Hajj. This means that the Shari’ah advice would be to pay off the interest loan in full instead of spending the money on Hajj. But if a Muslim decides to go for Hajj it is permissible and acceptable, InShaAllah, while he is still under a sin that invokes a war from God and His Messenger as the Qur’an [2: 279] tells us. This is because each action is reckoned on its own.”