I do not think there is a grain of truth/shred of truth in his remarks.

[nohabloingleas]

I do not think there is a grain of truth/shred of truth in his remarks.

Are both options correct?

 

[Piscean]

Yes.

 

[jutfrank]

I feel that the sentence is trying to emphasise that there's no truth at all, in which case only 'shred' works well. We don't usually use 'grain of truth' in that negative emphatic way.

I think you mean this:

There's not a shred of truth in what he says.