This page contains all the information you need on how to have a romance with Morinth.
Gather Information on Morinth
In order to continue this very little romance is only possible if you she survived the suicide mission in Mass Effect 2. Once you’ve completed the mission, Priority: Citadel I , check your Private Terminal and you’ll find a message from Morinth informing you that she decided to leave the Normandy whilst it was dry docked at the end of Mass Effect 2.
Make your way down to Liara’s room and access the Broker Terminal to find some emails from Morinth to her which have never been seen. Now continue progressing the game and when you’re ready initiate the mission, Priority: Earth. Here you’ll find Morinth as an enemy which you’ll have to kill in order to progress.