Well it's a generic phrase and if you say it's not infringing their TM and the site does not look confusingly similar then I'd be inclined to write to them and point this out. If they had a TM, they would be obliged to protect it, so 'turning a blind eye' would be possible grounds for having the TM revoked. It is possible that the act of their offering to 'turn a blind eye' would jeopardise their case too, but I guess you'd have to check with a lawyer if you can stand the company of such people.
Personally I think they are trying it on, but what to do would depend on how much income you get out of it and how much time and expense you have put into the site.
I believe a certain large low-cost airline has in the past managed to convince themselves that they own the rights to any domain starting with the first word in your domain, and have aggressively objected to many domains beginning with that word. I wouldn't be surprised (if they wanted to go into this industry) if they felt the name should belong to them too. But their past actions have left a trail of similar threats discussed across the Web, and these discussions and the outcomes may be useful to you as pretty similar test cases. Of course the company after you is far less well known and would have a lot less clout with an adjudicator in my opinion.
