If the offer letter mentions that they will contact your previous company for verification, there is no issue since you have already left the job. You can simply explain that your personal reasons or problems have been resolved, and you have found a good opportunity, which is why you joined. It's better to be honest rather than hide the truth. Importantly, once you leave a company, the previous employer has no right to interfere, so there is no need to worry if they contact them for background verification.
You should now email them to confirm that you are currently employed in your new position, stating that they requested you to send this email. This will inform them that you are working as (designation).
Regards, Sachin Kadam
"One single truth is always better than a thousand lies..."